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'I'  e 


(f.  "^^'^, 


THE 


ELEMENTS  OF  ALGEBRA; 


DESIGNED 


FOR  THE  USE  OF  SCHOOLS. 


EEV.    J.    W.    COLENSO,    D.D. 


BISHOP   OF   NATAL. 


PART  I. 


FBOM   THE   THIRTEENTn   LONDON   EDITION. 


NEW    YORK: 

JOHN  F.  TROW,  PRINTER,  50  GREENE  STREET. 

1867. 


CAJORI 


V 


ADVERTISEMENT. 


In  this  Edition  (wliicli  is  stcreotyjped^  and  so  will 
be  secured  from  further  change)  the  Simpler  Parts, 
those,  namely,  suited  for  general  School  purposes 
and  required  for  the  attainment  of  an  ordinary  B.A. 
degree  in  the  University  Of  Cambridge,  are  printed 
separately  as  Part  I ;  to  which  is  appended  a  large 
collection  of  easy  Miscellaneous  Examples,  specially 
adapted  to  the  contents  of  this  Part,  and  supplying 
means  of  complete  Examination  in  them. 

It  will  be  seen  tliat  the  easiest  kinds  of  Simple 

Equations  and  Equation  Problems  are  in  this  Edition 

introduced  much  earlier  than  is   usual  in  Treatijses 

,  on  Algebra:  but  there  can  be  no   reason   why  this 

^^fcanch  of  the  subject,  which  is  so  interesting  to  most 

^^Rudents,  and  gives  them  some  idea  of  the  practical 

il^)plications  of  the  Science,  should  not  be  brought 

forward  as  soon  as  possible. 

Part  II  is  also  published,  and  contains  the  higher 
parts  of  the  Subject,  with  such  additional  remarks  on 


IV  ADVKRTISEMENT. 

the  earlier  portions  as  M'ill  suit  the  wants  of  more 
advanced  and  promising  Students, -and  with  a  similar 
Appendix  of  more  difficult  Miscellaneous  Examples 
and  Equation  Papers.  This  Part  may  be  begun  as 
soon  as  the  Student,  having  thoroughly  mastered 
Part  I,  has  entered  upon  the  Miscellaneous  Examples 
at  the  end  of  it. 

Fomcett  St.  Mart/y  Nov.  1, 1849. 


TABLE  OF  CONTENTS. 


[^nxr.                                                                                                                       Page 
I.    Definitions 1 

II.    Addition 7 

Subtraction 8 

Use  of  Brackets 9 

Literal  Coefficients U 

Multiplication 13 

Division 17 

Jiesolution  into  Factors -  ....  22 

III.  Simple  Equations 24 

Problems 29 

IV.  Involution 33 

Square  of  Multiiiomial 34 

Evolution 36 

Of  square  root 38 

Of  ditto  in  numbers 40 

Of  cube  root 43 

Of  ditto  in  numbers 46 

Greatest  Common  Measure 48 

Least  Common  Multiple 57 

VI.   Algebraical  Fractions. 59 

Certain  Properties  of  Fractions G  8 

t                Remark  on  the  meaning  of  sign  = 69 
Ditto         .         .         .           sig7i  oc 70 

VII.   Simple  Equations  continued 71 

Problems ; 73 

Simultaneous  Equations  of  two  u  iknowns 81 

Ditto  of  three  unknoions 83 

Problems 84 


Vl  TABLE   OF  CONTENTS. 

Chap.  Pago 

VIII.  Theory  OP  Indices 87 

Surds 92 

IX.  Quadratic  Equations 99 

Simultaneous  Quadratic  Equations 105 

Problems 107 

Indeterminate  Equations Ill 

X.  Arithmetical  Progression 115 

Geometrical  Progression 116 

IIarmonical  Progression 121 

Arithmetic^  Geometric^  and  Harmonic  Means 123 

XI.    Ratio 124 

Proportion 126 

Variation 131 

XII.   Permutations 134 

Combinations 137 

Xlli.  Binomial  Theorem 140 

XIV.  Notation 148 

Decimals 151 

Interest,  &c 154 

Miscellaneous  Examples. 

Answers  to  the  Examples. 


ALGEBRA. 

PART  I. 


CHAPTER  I. 

DEFINITIONS. 

1.  Algebra  is  the  science  which  reasons  about 
quantities  by  means  of  letters  of  the  Alphabet,  and 
certain  signs  and  symbols,  which  are  employed  to  rep- 
resent both  the  quantities  themselves,  and  the  man- 
ner in  which  they  are  connected  with  others. 

Thus  we  might  put  a  to  represent  7,  and  then  twice  a  would 
represent  14 ;  or  we  might  put  a  to  represent  3,  and  then  twice  a 
would  represent  6,  tliree  times  a^  9,  &c. 

2.  The  sign  ==  {eqical)  denotes  that  the  quantities 
between  which  it  stands  are  equal  to  one  another. 

Thus,  if  (X  =  17,  then  txoice  a  =  34. 

3.  The  sign  .*.  stands  for  then  or  therefore^  and  •.•  for 
siJice  or  because, 

4:.  The  sign  +  (plus)  denotes  that  the  quantity  be- 
fore which  it  stands  is  added,  and  the  sign  -  (minus) 
that  the  quantity  before  which  it  stands  is  subtracted. 
Thus    5  +  3  =  8,  5  -  3  =  2 ;  and  if  cj  =  3  and  5  =  4, 
then     a  +  5  =  3  +  4  =  7,  «  +  J  +  2  =  3  +  4  +  2  =  9, 
10 -a  =  10 -3  =  7,  10-^-5  =  10-3-4=7-4  =  3. 

The  sign  ^  is  used  to  denote  that  the  less  of  two 
quantities  is  taken  from  the  greater,  when  it  is  not 
known  which  is  the  greater. 

Thus  a^h  denotes  the  difference  between  a  and  h. 

1 


5v      :  .     I>EFINITIONS. 

5.  All  quantities  before  which  4-  stands  are  called 
positive^  and  all  before  which— stands  are  called  nega- 
tive quantities. 

If  neither  +  nor  -  stand  before  a  quantity,  +  is  un- 
derstood, and  the  quantity  is  positive ;  thus  a  means 
+  a. 

6.  The  sign  x  {into)  denotes  that  the  quantities  be- 
tween which  it  stands  are  to  be  multiplied  together; 
but  very  often  a  full-point  is  used  instead  of  x ,  or, 
still  more  commonly,  one  quantity  is  placed  close  after 
the  other  without  any  sign  between  them. 

Thus  a  X  5j  (X .  5j  and  db  mean  all  the  same  thing,  viz.  a  multi- 
plied by  h ;   and,  therefore,   if  a  =  3  and   5  =  4,  we  shall  have 
ab  -  12,  6a  =  15,  6ah  =  60  ;  and  if  also  c  =  5,  tZ  =  0,  then 
Aah  +  3ac  +  4:d-2b  +  2alc-Zabcd  =  48  +  45  +  0-8  +  120-0 
=  213  -  8  =  205.      • 

7.  The  number,  whether  positive  or  negative,  pre- 
fixed to  any  algebraical  quantity  is  called  its  coeffi- 
cient ;  thus  3  is  the  coefficient  of  3a,  -  7  of  -  7aa?,  &c. 

If  no  number  is  expressed^  the  coefficient  is  under- 
stood, being  1,  since  a  means  once  a. 
Ex.  1. 

If  a  =  6,  5  =  5,  c  =  4j  tZ  =  3,  e  =  2j  /=  1,  and  g  =  0^  find  the 
numerical  values  of  the  following  expressions  ; 

\,  a  +  25+3c+4cZ+3e+2/+  ^.       2.  2a  +  5  -  3c  +  4cZ  -  5/+  6^. 

3.  35  -  4a  -  6c  +  7^  +  2e  -  Ag.    4.  -  3a  +  25  +  3c  -  2e  +/. 

5,  db  +  55c  -  Me  +  bfg.  6.  Aag  -  35/*+  4cc  -  ad. 

7.  -  3a5  -  2ac  +  45c  -  a5c.  8.  bob  -  8ac  +  \6cde  -  \Aaef, 

9.  33a5-19ccZ  +  22a5^-13c^4/:  \0.  abcd-2bcde  ^Zcdef-^efg, 

8.  The  sign  ~  (by)  denotes  that  the  quantity  which 
stands  lefore  it  is  to  be  divided  by  that  V!\\iQh.  follows 
it;  but,  most  frequently,  to  express  division,  the  quan- 
tity to  be  divided  is  placed  over  the  other  with  a  line 
between  them,  in  the  form  of  a  fraction. 


DEFINITIONS.  3 


a 


Thus  a-^h  and  j  dcnotCj  either  of  thenij  a  divided  by  b  ;  and 
if  <i  =  2,  5  =  3,  then 

5a_10  _5     Za  +  2h  _  6  +  6  _  1?  _  o 
25  ~  T  ~  3'     2b -a   "  6^  ~  4  ~    * 

9.  When  any  quantity  is  multiplied  by  itself  any 
number  of  times,  the  product  is  called  a  power  of  the 
quantity,  and  is  briefly  expressed  by  writing  down  the 
quantity,  with  a  small  figure  above  it  to  the  right 
denoting  the  number  of  times  it  is  repeated. 

Thus,  a^  stands  for  ^  x  a  x  ^  x  a  x  a,  3a*5Vc?  for  Zaaaabbbccd, 

The  small  figure  in  any  case  is  called  the  index  of 
the  corresponding  power. 

Thus,  a  (which  means  a^)  is  the  j^r^^  power  of  a^ 

a^ the  second    .     .    or  square  of  a^ 

a^ the  third      .     .    or  cube  of  a, 

a* the /owr^/^  power  of  «,  &c.  «S:c., 

and  the  small  figures,  ^,  ^  *,  &c.,  are  the  indices  of  the  second, 

third,  fourth,  &c.,  powers  of  a  respectivelj. 

Hence,  if  a  =  2,     a*  =  2  x  2  x  2  x  2  =  16, 

if  a  =  3,     «*  =  3  X  3  X  3  =  27,  ♦ 

if  a  =  1,    a^  =  1,    a'  =  1,  a*  =  1,  &c. 

Ex.  2, 

_^If  <x  =  1,  5  =  3,  c  =  5,  and  tZ  =  0,  find  the  values  of 

"      ,    25      3c     5a     2a +  b  n    ^^  +  25     25  +  oc     ^ab  -  c 

1.    — —  +   — h ,  Z»    ~ ■ = +    , 

a       l)        c  c  0  ia  a 

«    ah  +  2bc  +  Zed     2abc  -  4ad  +  2>ac      Zabc  +  6ac  +  6ab  -  35c 

6, ;^^ ^^ ^ +       . 


2^  +  35  Zab-2ad  6c -2b 

4.  a«  +  25^  +  3c^  +  M\  5.  Za'b  +  2h''c-2a''c  +  Wd. 

6.  a'  -  Za'^c  +  Zac''  -  c\  7.  «*  -  461^5  +  6a''¥  -  4a5^  +  5*. 

8.  Aalc^-Za^c  .  ^~.  9.  ^^'^^  .  -?^  _  «-^i!l£!. 
2a  +  5  +  c  Za^  a  +  b'  55' 

10    ^^!^1iJl      1  +  ^°c^      4a  +  5°  +  5'c'^     a=  +  2a5  +  5' 
*    'a^  +  5^  ""    a^  +  c'   "^       5*-'  +  c'        ""  "5^-  25c  +  c^ ' 


4  DEFmiTIONS. 

10.  The  square  root  of  a  quantity  is  that  quantity 
whose  square  power  is  equal  to  the  given  quantity. 

Thus  the  square  root  of  9  is '3,  since  3'  =  9  ;  the  square  root  of 
a"  is  a^  of  64  is  8. 

So  also  the  cube^  fourth^  &c.  root  of  a  quantity  is 
that  quantity  whose  cube,  fourth,  &c.  power  is  equal 
to  the  given  one. 

The  symbol  used  to  denote  a  root  is  V  (a  corruption 
of  7*5  the  first  letter  of  the  word  radix\  which,  with 
the  proper  index  on  the  left  side  of  it,  is  set  before 
the  quantity  whose  root  is  expressed. 

Thus,  V^'  =  «3  VG4  =  4,  V3125  =  5,  \Jl  =  1,  VI  =  1,  &c: 

The  index,  however,  is  generally  omitted  in  denoting 
the  square  root ;  thus  \/x  is  written  instead  of  y  x. 
Find  the  values  of  Ex.  3. 

1.  V4  +  2V25  +  3V49-VC4.         2.  3vl6-4v36  +  2V0-V^1- 

3.  V8  +  2  V125  -  4  VI  +  VG4.  4.  VI  +  3  V16  -  2  V32  +  3  VI. 
If  a  =  25,  5  =  9,  c  =  4,  fZ  =  1,  find  the  values  of 

5.  ^a  +  2^1  +  3Vc  +  4V^.  6.  ViS  +  ^1%  +  VISc- V253. 

7.  Z^a  +  2V46'-4V9^+  ^/l6d.     8.  V5^+2 V3&- V2c  +4 V^- 

9.  v«'  -  2  yh'  +  3  yc'  -  4  V^.     10.   V^+  3 V^-4V^^""+  Vc^». 

11.  Algebraical  quantities  are  said  to  be  liJce  or 
unlike,  according  as  they  contain  the  sa77ie  or  different 
combinations  of  letters. 

Thus  a  and  5a,  ^^a^h  and  la^l^  Za'^lc  and  -a'5(?,  are  pairs  of 
like  quantities  ;  a^  and  a"^^  Zah  and  -  7a,  Za^h  and  3a5^,  of  unlike 
quantities. 

12.  Brackets^{\  H?  []?  ^^^  employed  to  show  that 
all  the  quantities  within  them  are  to  be  treated  as 
though  forming  but  one  quantity.  It  is  of  great  im- 
portance to  notice  carefully  the  effect  of  using  them. 

Thus  rt-(&-c)  is  not  the  same  as  «-5-c;  for,  in  this  last, 
both  h  and  c  arc  subtracted,  whereas  in  the  former  it  is  the  quan- 
tity, J  -  c,  which  is  subtracted. 


DEFINITIONS.  0 

Hence,  if  a  =  4,  5  =  3,  c  =  1,  we  have 

a-.&-o  =  4-3-l  =  0,  a-(b-c)  =  4:'-2=^2; 

2a-35  +  2c.-8-9  +  2=I,  2a-{^b  +  2c)  =  S-U  =  ^Z; 

2a+  5-c  =  8  +  3-1  =  10,2(a+5)-c  =  14-1  -  13,  2  (a  +  5-c)  =12. 

Sometimes,  instead  of  brackets,  a  line  is  used,  called 
a  vinculum^  and  drawn  above  the  quantities  tliat  are 
connected;  thus  a—h—c  is  the  same  as  a—ip—c). 

The  line,  which  sei^arates  the  num"^  and  den'^  of  a 
fraction,  is  also  a  species  of  vinculum,  corresponding, 
in  fact,  in  Division  to  the  bracket  in  Multiplication. 

Thus Y —  implies  that  the  wJiole  quantity  a  +  5- c  is  to  be 

divided  by  4,  and  might  have  been  written  |  (<^  +  5  -  c). 

Ex.  4. 

Ifa  =  0,  ?^  =  2,  c  =  4,  ^  =  G,  find  the  values  of 

1.  3a  +  (25-c)'^+  |c^-(2a  +  3&)}  +  {?>c-(2a-\-Zl)}\ 

2.  Zl)^-{2c-dy  +  {35-(2c-cZ)}^-{3&-(2c-^'^}. 

3.  2  ^d-h  +  3  v3^  +  2c-l  +  4^^  +  h  +  2c  +  d. 


4.  3  V25=^-  «  +  2  V&'  +  c'^  +  7-V2  (5  +  cf--{h  +  cZ)^ 

5.  {a^(J)  +  cy-d]  {(a  +  'by  +  {d--cy]  |(^  +  5  +  c)'»-cZ}. 

If  a=l,  5=2,  c=3,  tZ=4,  shew  that  the  numerical  values  are  equal 

6.  Of(?>  +  c  +  ^)(5  +  c-6Z)  (h-^d-c)  (c  +  d^h) 

and  of45V-  ■[^=*-(52  +  c')p. 

7.  Of{^-(c-5  +  «)}  {(^  +  c)-(<^  +  «)K 

and  of  d^  -  (c^  +  ¥)  +  a^  +  2  (dc  -  ad). 

8.  0£{(h  +  c)-(d-a)}^+{(c  +  d)^(l>-ayf'+\{h  +  d)-(c-d)\''  + 

(b  +  c  +  d-ay,  and  of  4  (c^'*  +  5^+0^  +  tZ^). 

9.  Of  K«  +  ^-M)f  {(«+c+^-5}  \c-(d-a-iy  (h^c  +  d-a), 

and  of  4  (a^  +  dcy-{(a'  +  d'')-h^  +  c'')\\ 
10.  OfcZ^-(2^-c)c+  |2(J-c)  +  &}  l)-{2(d-c  +  h)-a\  a, 
andof  {((Z-a)-(c-5)}^ 

13.  Those  parts  of  an  expression,  which  are  con- 
nected by  the  signs  +  or  — ,  that  is,  which  are  connect- 
ed by  J.6Z(^^i^^(9^^/ or /S'?^5^rac^i6>^i,  are  called  its  terms.,  and 
the  expression  itself  is  said  to  be  simple  or  com/pound^ 
according  as  it  contains  one  or  more  terms. 


6  DEFINITIONS. 

Thus  a',  2abj  and  -35'*,  are  each  simple  quantities,  and 
a^  +  2db  -  ^h^  is  a  compound  quantity,  whose  terms  are  a^,  +  2a&, 
and-3&^ 

Those  parts  of  an  expression  which  are  connected 
by  Multiplication  are  called  li'^  factors. 

Thus  the  factors  of  a^  are  a  and  a,  those  of  2ab  are  2,  a,  and  6, 
those  of  -  35*  are  -  3,  5,  and  5,  or,  as  we  should  rather  say,  -  3 
and  5*,  it  not  being  usual  (except  where  specially  required  for  any 
purpose)  to  break  up  a  power  into  its  elementary  factors.  Of 
course  we  might  include  1  as  a  factor  in  each  case ;  thus,  since 
a''  =  1  X  a^^  the  factors  of  a"^  are  1  and  a*,  and  so  of  the  rest :  and 
this  will  be  sometimes  required,  as  will  be  seen  hereafter,  but  for 
the  present  need  not  be  attended  to. 

It  is  very  necessary  that  the  student  should  learn  at 
once  to  distinguish  well  between  terms  and  factors. 

Thus  2a  +  J)-c  is  a  compound  quantity  of  three  terms,  2a,  5, 
and  -c;  2  (a  +  'b)-c  is  one  of  two  terms  only,  2(a  +  b)  and  -  c, 
of  which  the  former,  2  (a  +  5),  consists  o^ two  factors,  2  and  «  +  5, 
the  factor,  a  +  d,  being  itself  a  compound  quantity  of  two  terms ; 
and  60  also  2  (a  +  h  -  c)  \s  a  simple  quantity  or  single  term,  of 
two  factors,  2  and  a  +  h-c,of  which  the  latter  is  itself  a  com- 
pound quantity  of  three  terms. 

Let  it  be  observed  then  that  terms  are  the  quantities 
which  make  np  an  expression  by  way  of  Addition  or 
Suhtraction,  factors,  by  way  o{  Multiplication. 

It  may  be  also  noticed,  that  it  is  immaterial  in  what 
order  either  the  terms  or  the  factors  of  a  quantity  are 
arranged.  It  is  usual,  however,  to  arrange  quantities, 
as  much  as  possible,  in  the  order  of  the  alphabet. 

Thus  a~2b  +  Zc  is  the  same  quantity  as  -  25  +  a  +  3c,  or 
3c  -  25  +  a,  &c.,  and  ahc  is  the  same  as  hac  or  hca  ;  but  we  should 
prefer  to  write  a  -  25  +  3c,  and  a5c,  unless  there  were  some  reason, 
in  any  case,  for  arranging  otherwise. 

A  quantity  of  07ie  term  is  called  a  monomial,  of  two 
terms,  a  hinomial,  of  three,  a  tri7iomial,  &c.,  and,  gen- 
erally, of  more  than  tivo  terms,  a  midtinomial. 


CHAPTEE   II. 

ADDITION,   SUBTRACTION,    MULTIPLICATION,   DIVISION. 

14.  To  add  like  algebraical  quantities,  add  sepa- 
rately the  positive  and  negative  coefficients  ;  take  the 
difference  of  these  two  sums,  prefix  the  sign  of  the 
greater,  and  annex  the  common  letters. 
Ex.1.    Za        Ex.  2. -125c        Ex.  3.    2c«       Ex.4.    Za''-^2l^ 

-2a  Zlc  -7c^  -Sa'  +  4l^ 

5a  -85c  10c'  5a* -65* 

6a  55c  4c*  7a* +  35* 


I 


7a  -85c  4c*  11a*     * 

In  the  last  example  the  star  is  used  to  indicate  that  the  terms  in- 
volving 5*  destroy  one  another. 

If  the  quantities  are  itnliJce^  we  must  add  any  that 
are  like  by  the  preceding  rule,  and  write  down  the 
others  with  their  proper  signs. 

Ex.  5.    2a+35-4c        Ex.  6.    x-2y-^'^z        Ex.  7.  2a+c-^cl 
-3a+45-  c  -2x+oy-4z  -h+a+e 

4a +75+ 7c  Sx-57/-5z  +    c-d 

a-  5- 4c  ^ -^  y  -3a-e  -/ 

-5a + 25-  6c  2y  +  2z  -  2c + 2d  -2e 

-a +155- 8c  Zx-  y-4z  -5  +2d-2e-f 

Find  the  sum  of  Ex.  5, 

1.  7a-35+4c-2(Z+7. -8a  +  45-6c+2^Z-ll,  13a  +  35-5c+4^-4, 
2a-5+c+ll,  a+2d-Z. 

2.  2x-Sy  +  4:Z-4,  x  +  2y-Sz^  -Sx  +  2y  -5z  +  7,  ix-y  +2^-3, 
9a;-10?/  +  ll2-12,  x^y^z, 

3.  2a*+a5  +  35*,  3a*-4a5  +  25*,  3a*  +  3a5-5*,  12a*  -  14a5  -  75', 
3a*-12a5  +  175*. 

4.  ax-4J)y  +  3cz,  lZax-%y  +  7cZj -5ax+7hy-14:cz^2ax-'by  +  czy 
"llax+lZdy  -4cz, 

5.  20a;^  +  20x^y  -  3xy^  +  Uy^,  -  17a;^  +  Ux'^y  -  Uxif  -  Zy\  14a^ 
f  +17a;*j/  +  15a'y*-5?/',  -  12.c'- 13aj*y  -  14a:2/*- 5^',  I2x^y  +  Zy\ 


8  SUBTRACTION. 

6.  2iB'  -  Zx7j  -  4i/^,  Zxz  +  22/^-  s^  ic'-  2^^  +  62*,  Sxj/  -  6iC2  -  Zx\ 
Zxz-2z''+6yz,  4y''-Zyz+2x\ 

7.  cc'-  Zo^x'^  +  Za^'x-a^,  4x^-  5ax^  +  6a''x  -  15a\  ZxUAax-^  2a''x 
+  6a^  -  17aj'  +  19aa;^  -  15a=a;  +  8a^  -  ISaa;'^  -  27a'a;  +  lSa\ 

8.  «'-  2a6'-  ac^^^-  a=Z>  +  2a'(j  +  2ahc,  -  a'^^  +  i^-  26c=+2aZ;=+2aZ)c 
+  h\  -2a^c  -  h^c+c"  +  2abc  +  ac^  +  25c^ 

9.  Zx^  +  2^/^  +  2!^  +  8^2^  2/'  +  3a;V  +  2icy^  +  s'  -  SSaj'^g,  jc»  +  2xyz 
-\-4x^y  +  l^ic'^g  -  Oy'^z  +  6yz^,  2x^-  Zy^+  4xyz-(jxy^,^  4y^  -  z^+  6x'z 

-  16xyz  +  oy'^z  -  142/2^,  ^x'^z  -  15xyz  +  Axy^  -  Ix^y  +  G?/^2. 

10.  aj'*+  Sxy^-  xz^+  xhj+x^z^  Zx^y"^ ^Zx'^z'^  +  Zxy^z  —  Zxyz^  -  Ox^yz^ 
-x^y+  y^-  yz^-  Zx'^y'^-^-  Zx^yz,  -  Zxy^—  Zxyz^—  Zy^z  +  Zy'^z^-  Gxy'z, 

-  x^z  +  Zy^z  +  z*  +  Zx^yz  -  Zx^z^,  Zxy^z  +  xz^  -  Zy'^z^  +  yz^  +  Gxyz^. 

15.  To  subtract  algebraical  quantities,  change  their 
signs  and  proceed  as  in  Addition. 

Thus,  if  we  take  h  from  ^,  the  result  will  be  a  -  5 ; 
but,  if  we  take  h-o  from  a,  the  result  will  be  greater 
by  G  than  the  former,  since  the  quantity  now  to  be 
subtracted  is  less  by  c  than  in  the  former  case ;  hence 
the  result  required  will  be  a-h+Cj  which  is  therefore 
the  value  of  ^  -  (5  -  c),  so  that  the  quantities  5,  -  c^ 
when  subtracted,  become  -  5,  +  <?,  respectively. 

Or  we  may  reason  otherwise,  as  follows : 

(i)  Since  a=^  a -!}-[- h^ii  we  subtract  +  ^  from  a,  the 
result  is  a  -  5,  the  same  as  if  we  add  -  5  to  it ; 

(ii)  Since  (^  =  a  +  &  -  5,  if  we  subtract  -h  from  a^  the 
result  is  a-\-i^  the  same  as  if  we  add  +5  to  it. 

Thus  if  a  person  possesses  a  pounds  and  owes  h  pounds,  his 
money  in  hand  may  be  represented  by  +  «  pounds,  and  his  debt 
by  -  &  pounds,  so  that  he  may  be  said  to  possess  +  a  and  -  h 
pounds,  or,  in  one  sum,  a-l)  pounds.  Now  if  we  ttihtract  or 
annul  his  debt,  that  is,  if  we  take  away  his  negative  property, 

-  2>  pounds,  he  will  possess  the  whole  positive  property,  +  a  pounds, 
the  same  as  if  we  give  him  +  h  pounds,  to  jmy  his  debt  with. 

There  will  often,  however,  be  no  need  formall}^  to  apply  the  above 
Tiiloof  changing  signs,  since  the  difference  may  be  obtained  at  once, 
by  taking  that  of  the  coefficients  and  annexing  the  common  letters. 


I 


USE   OF  BRACKETS.  9 

Thus,  in  Ex.  1,  we  may  say,  at  once,  3a;  from  hx  leatcb  ^x 
y  from  ly  leaves  6y,  -Azfrom  -8s  leaves  -4s ;  though  of  course, 
if  we  chose  to  apply  the  Rule  {change  the  sign  of  the  quantity  to 
he  subtracted  and  proceed  as  in  Addition)  it  would  equally  be 
true  that  -Zx  added  to  +5a;,  -y  to  +  ly^  +  4s  to-  8s,  would  pro- 
duce respectively,  +  2^,  +  6y,  -4s,  as  before. 

Ex.  1.  Ex.  2.  Ex.  3. 

From  5a;  +  7y  -  8s         5a;^  -  2xy  +  Zy'^         -  3a^  +  Adb  -  55^ 
take    3a;  +    y  -  4s      -  4a;^  -  2a;y  +  7y^  -  7^^  -f  35'  -  2c^ 

Ans.  2a;  +  Gy-4s         9a;=*  ^^'^  4a'  +  4«2> - 8f'~72c' 

Ex.  6. 

1.  From  2<^- 25  +  c  take  a  +  5  -  2c;. 

2.  From  2a;'  -  3a;y  +  y-  take  4a;'  +  4a;y  -  2y^. 

3.  From  bax  -  75y  +  cs  take  ax  +  25?/  -  cz. 

4.  From  7a;'  -  2a;  +  4  take  2a;'  +  3a;  - 1. 

5.  From  8ii'-2a  +  65'  -5^5  +  5c'  -  35c  +  2 
take    a'  +  a  +  25'  +  2a5  +  3c'  +  35c  +  2. 

G.  From  2a;^  -  ^x'^y  -  3?/'  +  6  -  2a;'  -  3a;2/'  -  14i/» 
take  3a;^  +  2x'^y  -y"^-  3a;?/'  +  a;'  - 10?/^ 

7.  From  5a;'  +  (jxy-  4i/'  -  12a;s  -  lyz  -5s' 
take  2a;'  -  3?/'  +  4a;s  -  5s'*  +  Q>yz  -  7xy. 

8.  From  3a;'  +  ^xy-y""  take  -a;' -  3a;?/ +  3?/',  and  3a;'  +  Axy '-by\ 

9.  From  a*-2ci=5+3a'5'-4a5'  +  55*  take  2a5^-.3a'5'-^4a^5-5a*, 
and  3a*-2a^5  +  6a'5'-2a5'  +  35*. 

10.  From  a''  -  4a'5'  -  8a'5^-  17a5*  - 125'   take  a^  -  2a*5  -  3a»5', 
2a*5-4a'5'-6a'5^  3«^5' -  Cc^'5'-9a5^  and  4a'5^-8t*5*-125*. 

16.  Since  the  sign  +  or-,  preceding  a  bracket,  will 
imply  (12)  that  the  whole  included  quantity  is  to  be 
added  or  subtracted,  if  we  wish  to  remove  the  bracket, 
we  must  actually  perform  the  operation  indicated  by 
means  of  it,  i.  e,  we  must  add  or  subtract  the  quantity 
in  question.  Of  course,  in  the  case  of  +  preceding  it, 
this  amounts  to  no  more  than  merely  setting  down 
the  included  terms  with  their  proper  signs,  because, 
when  a  quantity  is  added,  the  signs  of  its  terms  are 
not  altered ;  but  in  the  case  of  -  preceding  a  bracket, 
1^ 


10  USE   OF  BRACKETS. 

we  shall  have  to  change  the  signs  of  all  the  included 
terms,  since  they  are  all  to  be  subtracted. 
Thus  +  (a-¥l-c)=     a  +  h-c,     (a^  -  2a5  -  &')  =     a^  -  2ab  -  J' ; 
but     -  («  +  5  -  o)  =  -  a  -  5  +  c,  -  (a'  -  2ab  -  J«)  =  -  a'  +  2a&  +  &' : 
BO  also,  in  the  case  of  a  double  bracket,  we  have 

Za-  {a- 3c)  +  (25 - c)    = 
3a  -    a  +  Zc  +  2b-c    =2a  +  21?  +  2c. 
The  same  remark  applies  also  to  the  case  of  a  frac- 
tion with  a  num''  of  more  than  one  term,  whenever  the 
line  separating  its  num'  and  den',  and  which  (12)  is  a 
species  of  vinculum,  is  removed  by  any  process. 

Thus ^ [or-i(a+5-c)]  =  ----  +  - lov-la-ll  +  |c] ; 

and  -  |(a  -  5)j  when  multiplied  by  2,  becomes  -(a-l)^  or  -  a  +  I. 

Ex.  7. 
Reduce  to  their  simplest  forms  : 

1.  {a-x)-(2x-a)-i2-2a)  +  (3-2^)  -  (1-aj). 

2.  {a'-2a''c  +  Zac'')-(a'c-2a^  +  2ac')  +  (a'-ac^-a^c). 

3.  (2^''  -.  2y^  -  z')  -  (3if  +  2x^  -  z")  -  (3s^  -  2^/'  -  a;'). 

4.  (a;'  +  ax^^a'x)-'{y^-ly'  +  5^)  +  («'  +  <^^''  +  c'2)  -  (a;' -  2/'  +  e') 

+  {ax^  +  5?/^  +  cz')  -  (a'x  -  Vy  +  c'e). 

5.  a'  -  (&'  -r)  -  {5^  -  (c^ - a^)f  +  jc' -  (&^ -«')}. 

6.  |2«^-(3a&-5^)}-{6j^-(4a&  +  5')}  +  \2l''-{ci?-ab)\. 
r.  ja;'  +  2/'  -  (Sa^V  +  Sa;^/') }  -  K^'  -  Sx^)  -  (Sa-y'  -  j/') }. 

8.  {2aj~(32/-2)}  -  \y  +  (2a;-0)}  +  {33-(aj-22/)}  -  |2r-(7y-2)}. 

9.  l-{l-(l-4aj)f  +  {2a;-(3-5a;)}-j2-(-4  +  5a;)f. 

10.  |2a-(35  +  c-2^}-|(2a-3&)  +  (c-2(Q}  + -;2a-(3J+c)-2if 
-|(2<^-35  +  c)-2^f. 

17.  It  is  often  necessary  not  only  to  break  up,  or 
resolve^  quantities  contained  in  brackets,  but  also  to 
form  such  quantities,  that  is,  to  take  up  in  a  bracket 
any  given  terms  of  an  expression.  Now,  in  doing  this, 
it  should  be  noticed  tliat,  whatever  terra  we  choose 
to  set  ^^  first  term  within  the  bracket,  the  sign  of  that 
term  will  have  to  be  placed  heforeX\\Q  bracket,  and  this 


USE   OF   BRACKETS.  11 

Sign  will  of  course  affect  all  the  terms  we  may  place 
within  the  bracket.  If,  then,  this  sign  should  be  (+), 
the  other  terms  may  be  set  down  at  once  within  the 
bracket  with  their  proper  signs ;  but  if  it  should  be 
(— ),  we  shall  have  to  change  the  signs  of  all  these  other 
terms,  and  then  set  them  w^ithin  the  bracket :  for  the 
sign  (— ),  which  precedes  the  bracket,  will  influence 
all  these  signs,  and  have  really  the  eff'ect  of  correcting^ 
as  it  were,  the  changes  we  have  made,  and  will,  in 
fact,  cause  the  original  signs  to  reappear,  whenever 
we  choose  to  resolve  the  bracket  again. 

Thus  +  a-h-c,  collected  in  a  bracket  with  +  «  as  first  term, 
will  be  +  {a  -h-c);  but,  with  -  5  as  first  term,  -  (5  -  «  +  c),  and 
with  - c  as  first  term,  -(c-a  +  b)  ;  and  now,  if  we  resolve  again 
these  last  two  brackets,  the  sign  (-),  preceding  each  of  them,  will 
correct  the  changes  Ave  have  made,  and  the  quantities  will  be  re- 
produced, as  at  first,  -h  +  a-Cy-c+a-K 

So  also  we  might  use  an  inner  bracket,  and  write  the  quantity 
+  \(a-h)-c},  or  -i-ja-(J+c)},  or  -\(h-a)+c}^  or  -{b-ia-c)},  &c 

Ex.  8. 

Express,  by  brackets,  taking  the  terms  (i)  Uco,  (ii)  three,  together, 

1.  2a-b-Sc+4d-2e+Sf,  2.  -&-3c+4^-26+3/+a. 

3. -3e+4^-2e+3/+2«-5.  4.  +4iZ-2^+8/+2a-5-3(j. 

5.  -2e+2f+2a-b-dc+U,  6.  3/+2a~5-3c+4(Z-2e. 

7 — 12.  Express  the  second  answer  in  each  of  the  above  by  using 
also  an  itmer  bracket,  including  in  it  the  latter  two  of  the  three 

terms  within  each  of  the  outer  brackets. 

•  - — _— . 

18.  We  have  spoken  hitherto  only  of  nwnerical 
coefficients ;  but,  in  fact,  when  a  quantity  is  composed 
of  two  or  more  factors,  any  one  of  them  is  a  coeffi- 
cient of  the  rest  taken  together,  that  is,  (as  the  word 
coefficient  implies)  makes  Ujp  with  them,  as  a  factor, 
the  quantity  in  question. 

Thus  in  Zabcx^  3  is,  as  before,  the  coefficient  of  abcx ;  but  8a  is 
also  the  coefficient  of  bcx,  dab  of  ex,  ax  of  'dbc,  &c. 


12  USE  OF  BRACKETS. 

Such  coefiicients  are  called  literal  coeflScients,  as 
involving  algebraical  letters  ;  and,  when  any  terms 
of  a  quantity  contain  some  common  factor,  a  bracket 
is  often  employed  to  collect  the  other  factors,  consid- 
ered as  its  literal  coefficients,  into  one  quantity, 
which  is  set  before  or  after  the  common  factor. 

Thus  we  have    seen    already  that  3a;  +  2x-x  =  4x,  that  is, 
=  (3  +  2  - 1)  a; ;  and  in  hke  manner,  ax  +  hx  -  x  =  (a  +  h  -  l)Xj 
2a  -  Aax  +  Gay  =  2a  (1  -  2i&  +  Zy),  (a  +  2h)  x"-  (25  -  c)  x"--  (2c  -  a)  ic* 
=  \{a^2l)  - (25 - c) - (2c-a)}  x""  =  {2a- c)  x\ 
Add  {a  -  2p)  x^        -  2x'  +  {2c  -  3?-)  x 
(2p  -^r  0L)x^  ^  {<i-V)x^  -X 

"(P  -  a)  cc*  -  (5  +  q)  x^-  {c  -  l)x        From  ax^      -hx^      a-   x 
-x'^         +  35a;^-  (c  -  2r)  x     take  -  px^      -  qx^      +  rx 

Ans.  (Zor-p-l)  x^  +  (5-2)  x"^  -  rx  Ans,(a  +p)x^-{b-q)x'^  +  ( l-7')x 

The  above  Answers  ma}^,  of  course,  be  expressed  difTerentlj,  by 
changing  the  order  of  the  terms  within  the  brackets ;  thus,  the 
second  might  have  been  written  (a-hp)  x^+  ($'-  5)  x^-  (r- 1)  x. 

On  the  other  hand,  when  a  bracket^  comes  in  this 
way  before  or  after  a  single  term  as  factor,  it  may  be 
resolved,  after  multiplying  each  term  of  the  quantity 
within  it  by  the  common  factor. 

Thus  a(b-x)-(a-y)h=  (ah  -  ax)  -  (ah  -  hy)  - 
ah  -  ax  -ah  -^hy  =  hy-ax  =  -  (ax  -  hy), 

Ex.  9. 

1.  Collect  coeff '  in  ax^-  hx'^-  ex-  hx^^-  c.r''-  dx  +  cx^-  dx^-  ^* 

2.  Add  together  ax  -hy,  x  +  y,  and  (a-l)x-(h  +  1) y, 

3.  Add  together  (a  +  c)  x""- 3  (a -h)xy  +  (h- c) 2/^  and 

(h  -c)x'^  +  2(a  +  h)  xy  +  (a- h) y"^, 

4.  Add  together  (a  +  h)x  +  (h  +  c)y  and.  (a-h)x-(h-c)y^ 

and  subtract  the  latter  from  the  former. 

5.  Add  together  (i)  the  first  two,  (ii)  the  last^  two,  and  (iii)  all 

four    together,  of   2  (a  +  h)  x  +  S  (h  +  c)y,  -Z  (a-h) x  + 
2(a-c)y,  -(2h+c)x  +  (a-2h)y,  and  (a  -  25)  j  -  (5  +  2r)^. 


MULTIPLICATION.  13 

6.  In  (5)  (i)  subtract  the   second  quantity  from  the  first,  and 

(ii)  the  fourth  from  the  third,  and  (iii)  add  the  two  results 
together. 

7.  In  (5)  (i)  subtract  the  third  from  the  first,  and  (ii)  the  fourth 

from  the  second,  and  (iii)  add  the  two  results  together. 

8.  In  (5)  (i)  subtract  the  fourth  from  the  first,  and  (ii)  the  third 

from  the  second,  and  (iii)  add  the  two  results  together. 


I 


19.  To  multiply  two  simple  algebraical  quantities 
together,  multiply  together  respectively  the  numerical 
coefficients  and  letters ;  and  then,  if  the  multiplier 
and  multiplicand  have  the  same  sign,  prefix  to  this 
product  the  sign  +,  \i  different  signs,  the  sign  ~, 

Thus,  la  X  4&  =  28a5,  -  2a  x  3c  =  -  6ac,  55  x  -  2c  =  -  105c, 
-  3<a^  X  -  55  =  15ct5. 

This  rule  for  determining  tlie  sign  of  the  product, 
viz.  that  like  signs  produce  +  and  imlike  ~,  may  be 
thus  deduced. 

Let  it  be  required  to  multij^ly  a-hhj  c-d. 
Here  {a-'b){c-d)  =(a-l)  x^  (writing  x  for  c-d)y 
=^  ax-hx 

=  a  {e-d)-'b{c-d) 
=  {ac  -  ad)  -  {ho  -  bd) 
=  ac-ad-hc  +  hd: 

in  which  result  we  see  that  the  product  of  +  ahj  +€ 
is  ac  (i.  e.  +  ac),  that  of  +  ahj  -d  is  -ad,  that  of -Z> 
by  +  c  is  -  Ic,  and  that  of  -  5  by  -dis  +  id. 

If  several  simple  quantities  are  to  be  multiplied  together, 
instead  of  multiplying  them  together  successively  by  the  above 
rule,  (thus  2a  x  -  35  x  -4c  =  -  6a5  x  -  4c  =^  24a5c),  it  will  be 
shorter  to  multiply  them  at  once  together,  and  then  prefix  to  this 
product  the  sign  +  or  -,  according  as  the  number  of  negative 
factors  is  even  or  odd. 


14  MULTIPLICATION. 

20.  The  powers  of  a  quantity  are  multiplied  toge- 
ther by  adding  the  indices. 

Thus  «*  X  a'  =  cb^+^  =  a^ ;  for  a^  =  a.a  ,a^  a'^  =  a.  a; 
.'.  a^  y  a"^  ==  a,  a .  a  ,a,a  =  a^  ]  and  so  in  other  cases. 

Hence 
^  3a"5  X  Aa'h^  x  -  2a'^5»=24a^ J«,   2ahc  x  Za^h'^c^  x  -  aJ'c  =  -  Ga*h'e\ 

21.  If  the  multiplier  or  multiplicand  consist  of  seve- 
ral terms,  each  term  of  the  latter  must  be  multiplied 
by  each  term  of  th,e  former,  and  the  sum  of  all  the 
products  taken  for  the  complete  product  of  the  two 
quantities. 

This  process  is  generally  conducted  as  in  the  following  Ex- 
amples. 

Ex.1.  Sx^    -    2x7j    +     42/'        Ex,  2.  -2a^&V    5a5»  -    71* 
2a^x  ~4a5 


Ex.  3.  a  +  5  Ex.  4.  a  +  J 

a  +  5  a-h 


a*  +  ah  a^  +  db 

+  db  +  h*  -  ab-V 


Sa'h' 

-20a'^J*  + 

28a&» 

Ex. 

5.  a- 

-I 

a- 

-I 

a'- 

-db 

• 

-db 

+  J» 

Ex.  ^.  x-^-  a  Ex.  7.  a;'  +  (<i  +  5)  a;  +  a5 

a;  +  5  a;  +  <5 


aj'  +  ex  a;'  +  (a  +  J)  a;'  +  a5  a; 

+  &a;  +  a5  +         c  a;'  +  (ac  +  5c)  a;  +  aJo 


^7W.  a;*  +  (a  +  5)  a;  +  a?)  a;'  +  (a+5+c)  a;'  +  (a5  +  ac+5c)  aj+aJ^ 

Ex.  8.  a;'  -  aaj'  +    Ja;   ~   c 


a;'^  -  ax*  +    5a;*  -    c  a;' 

+  m  a;*  -  aw  a;'  +  hmx^  -  on  x 

-\-  nx^  -  an  x^  -^hnx-cn 


Ans.  a;*  -  (a  -  m)  x*  +  (b~am  +  n)  x^-  (c-hm  +  an)  x^  -  {cm-bn)  x-cn 


MULTIPLICATION.  15 

Ex.  10. 

1.  Multiply  ax^y^  by  Ixtj ;   mx^  by  -  nx* ;   -  acx  by  -  2axy  \ 

ale  by  5c ;  -  abc  by  -  «c  ;  x^y  by  -  xy'^, 

2.  Multiply  x^-xy  +  2/^  by  aj,   and  a^  -  aa;  +  a;"  by  -aa;; 

a;'^  -  aa;  +  5  by  -  aJa; ;  a;^  -  Zx'^y  +  3a;2/^  -  y"^  by  ajy. 

3.  Multiply  2a  +  5  by  a  +  35,  and  2a  -  5  by  c  -  Zd, 

4.  Multiply  3a;  +  2y  by  2ai  +  3y,  and  Zdb  +  45''  by  2a5-  35\ 

5.  Multiply  a;'*  +  3;i;  -  2  by  a;+3,  and  aj^  -  4a;  +  3  by  a;- 2. 

6.  Multiply  a^  +  2a  -  1  by  a^  -  a  +  1,  and  by  a^  -  3a  - 1. 

7.  Multiply  27a;''  +  9a;V  +  3a;2/^  +  2/'  by  3a; -y. 

8.  Multiply  a*  -  2a'6  +  4a''6''-8a5^  +  165*  by  a  +  25. 

9.  Multiply  x''  +  2aa;  +  3a'  by  a;'-2aa;  +  a^ 

10.  Multiply  9a'  -  3a5  +  5'  -  6a- 25  +  4  by  Sa  +  5  +  2. 

11.  Multiply  x"^  +  y"^  +  s^  +  xy -xz  +  yz  hy  x-y  +  z. 

12.  Multiply  a^  +  2a'  +  2a  +  1  by  a^-2a'  +  2a-l. 

13.  Multiply  a'  +  45'  +  9c'  +  2a5  +  Sac- 65c  by  a -25 -3c. 

14.  Multiply  a*-2a^5  +  3a'5'-2a5^  +  5*  by  a'  +  2a5  +  5'. 

15.  Multiply  a;'-aa;  +  5  by  x-c,  and  by  a;'  +  ax-c, 

16.  Multiply  1-aa;  +  5a;' -co;'  by  1  +  x--x^. 

17.  Multiply  a  +  mx-nx'^  by  a-2;wa;  +  na;',  and  by  a  +  2nx-mx\ 

18.  Find  the  continued  product  of  ax-hy^  ax  +  cy,  and  ax-dy» 

19.  Find  the  continued  product  of  2a;-w,  2a;+^,  x+2mj  and  a;-27i. 

20.  Find  the  continued  product  of  a;'  +  aa>-5',  a;'  +  5a;-a'j  and  a;-(a  +  5). 

22.  The  student  should  notice  some  results  in  Mult", 
so  as  to  be  able  to  apply  them  when  similar  cases  occur, 
and  write  down  at  once  the  corresponding  products. 

Thus,  (21  Ex.  3.  5)  the  product  of  a  +  5  by  a  +  5,  or  the  square 
of  a  +  5,  is  a'  +  2a5  +  5',  and  the  square  of  a  -  5  is  a'  —  2a5  +  5' : 
by  remembering  these  results,  we  may  write  down  at  once  the 
square  of  any  other  binomial ;  thus, 

(x + yy  =  a;' + 2a;y + 2/',   (x  -  2)'=  a;'-  4a; + 4,   (2a; + 2/)'=  4a;'  +  4a;y  +  y\ 
(2aa;-35y)'  =  4a'a;'  -  12a5a;y  +  95'^/'. 

Again.  (Ex.  4)  the  product  of  a  +  5  by  a-  5  is  a'  -  5' : 
hence  we  have  (a;  +  2^)  x  (x-y)  =  a;'  - 2/',  (a;  +  2)  (a;-  2)  =  a;' - 4, 
(2aa;  +  Zhy)  (2ax  -  3hy)  =  4a'a;'  -  95'2/'. 

So  also,  (Ex.  6)  the  product  of  a;  +  a  by  a;  +  5  is  a;'+  (a  +  5)  a;  +  a5, 
where  the  coeff.  of  x  is  the  sum  of  the  two  latter  terras  of  the 


16  MULTIPLICATION. 

factors,  X  +  a,  x+h,  and  the  last  term,  +  a6,  is  their  product :  in 
like  manner,  we  shall  have 

(x  +  5)  (x  +  2)  =  x^  +  (5  +  2)x  +  10  =  x^  +  7x  +  10, 
■  (x-b)  (a;  +  2)  =  a;^  +  (2-5)a;-10  =  ic'-3a;-10, 
(x  +  2)  (x-  2)  (x  +  3)  (x-S)  =  (ic*-4)  (a;^-9) 

=  aj*-(9  +  4)aj'  +  36  =  a;*  -  Ux^  +  36, 
(a;  +  2)  (a;  -  3)  (a;  -  4)  (a;  +  5)  =  (a;^  -  a;  -  6)  (x^  +  a;  -  20) 
=(by  common  Mult")  ;?;*  -  27aj^  +  14a;  +  120. 

23.  Let  then  these  three  results,  ovformulcB^  be  noted: 

(i)(^±  IJ^a'  ±^ah-VV', 
or,  the  square  of  any  hinomial  =  the  sum  of  the  squares 
of  its  two  terms  together  with  twice  their  prodicct  : 

{\i)  {a  -\-l){a  --1))^  a" -V\ 
or,  the  product  of  the  sum  and  difference  of  any  tioo 
quantities  =  the  difference  of  their  squares  : 

(lii) {x  -{-  a)  {x  +  h)  =^ x^  +  {a  '\-h)  X -\- ah. 

24.  By  a  little  ingenuity,  however,  the  above  formulae  may  be 
still  more  extensively  applied  to  lighten  the  labor  of  Mulf* :  thus 

Ex.1.  {a-h  +  cy=:  {{a-h)  -^-cY^hy  (\)  (a -hy  +  2{a-h)c-^c^ 
=a?  ~  2ab  +  Z)'*  +  2ac  -  2bc  +  c* ;  or  we  might  have  written  it 
{a  -X  (h-c)\^j  or  {(p  +  c)  -  h}^,  &c.,  and  then  have  expanded  either 
of  these  by  (i),  obtaining,  of  course,  the  same  result  as  before : 
but  we  shall  give  a  better  method  hereafter  for  squaring  a  trino- 
mial ;  it  will  be  sufficient  to  have  noticed  this. 

Ex.  2.  (a^  -  ax  +  a;')  (a^  -  ax  -  x^)  =  by  (ii)  (a"^  -  axY  -  x* 

=  a*  -  2a^a;  +  a'x^  -  x*. 

Ex.  3.  (a^+ax  -  x"")  («'-  ax  -x^)={ (a""-  x^)+ax}  { (a""-  x^)  -ax\ 
«  (a"  -  x^y  -  a^x^  =  a*  -  2a''x'  +  x' -  a^x^  =  a^  -  Sa'a;'  +  x\ 

Note  that  the  formula  here  employed,  (a  +  h)  x  (a  -  h)  =  a"^-  t', 
may  be  always  applied,  whenever  it  is  seen  that  the  two  quantities 
to  be  multiplied  consist  of  terms,  which  differ  only  (some  of  them) 
in  sign,  by  taking  for  a  those  terms  which  are  found  iclth  their 
signs  unaltered  in  each  of  the  given  quantities,  and  the  others  f<.ir 
b:  thus,  in  Ex.  3,  a^  and  ~  a;^  appear  in  both  the  given  quantities, 
whereas  in  one  we  have  +  ax,  in  the  other  -  ax ',  hence  the  pro- 
duct required  is  (a^-x'^y-a^x\  as  above. 


DIVISION.  17 

Ex.  4.  (a^+  ax  +  x^)  {a^-ax+x"*)  =  (a^+  x^y-a^x^  =  a*  +  aV  +  ic'. 
Ex.  5.  (aVaa;-a;'.)  (^2-aa;+a;'')=c*'-(aawc')^=a*-aV+2aa;'-ic*. 
Ex.  6.  (a'^-^aj+a;')  (aa;+a;''-a')=a;*-(«^-aa;)'=a5*-a*+2a=a;-aV. 
Ex.  7.  (a  +  5  +  c  +  ^  (<^  +  &-C  -  ^)  =  (a  +  &)*  -  (c  +  cQ' 

=  a-  +  2a5  +  P-c^-2cd-d\ 
Ex.  8.  (a  +  25-Sc-^  (a-2I>  +  3c-^  =  (a-6r)^-(25-3(j)« 

=  a^  -2a^  +  d''-4:P  +  12Z^c-0c^ 

Ex.  11. 

1.  TV'rite  down  the  squares  of  a-x,  1  +  2aj^,  2(x'^  +  3,  3aj  -  4y, 

2.  Write  down  the  squares  of  3  +  2x,  2x  -  3y,  a^  -  3aaj,  Jo;^  -  cxy. 

3.  Write    down  the  product    of   (2a  +  1)  x  (2a  -  1),  (3aa;  + 1) 

x(Sax-d),  (x-1)  (x+1)  (x^  +  1). 

4.  Write  down  the   product  of  (x  +  Z)  (x  +  1),  (ajV  4)  (ic'-l), 

(a&-3)  (^h  +  2),  (2^aj-3&)  (2«aj-5). 

5.  Find  the  continued  product  ofx  +  a,x-a,x  +  2a,  and  x  -  2a, 

6.  Find  the  continued  product  of  mx^2ny^  mx-2ny,  mx-Zny^ 

and  mx  +  3?i2/. 

7.  Simplify  2>  (a-2xy  ■v2(a-2x)  (a^2x)  +  (^x-a)  (Zx+ay(2a-'Zxy, 

8.  Multiply  ic^  +  2ajy  +  2y''  by  a;'-2iC2/  +  %'?  and  2a=-3a&  +  l^ 

by  2a''  +  3aZ>  +  5^ 

9.  Multiply  a^h  ■¥  c  by  a  +  h-c,  by  a-h-r  c,  and  by  a  -  5  -  c. 

10.  Multiply  «  -  5  +  c  by  a  -  2>  -  c,  by  Z>  +  c  -  o^,  and  by  c  -  Z>  -  a. 

11.  Multiply  2a+h-3chj  2a -I  +  3o,  and  by  5  +  3c -2a. 

12.  Multiply  2a- 5-  3c  by  2a  +  &  +  3c,  and  by  5 -3c -2a. 

13.  Multipl}'-  a  -^-l)  +  c  -^^  d  by  a-l  +  c-d^  by  a-d-c  +  d, 

and  by  5  +  c  -  c?  -  a. 

14.  Multiply  a  -  2Z>  +  3c  +  (Z  by  a  +  2Z>  -  3c  +  ^,  by  25  -  a  +  3c  +  d?, 

and  by  a  -f  25  +  3c  -  ^. 


25.  To  divide  one  simple  algebraical  quantity  by 
another,  divide  respectively  the  coefBcient  and  letters 
of  the  dividend  by  those  of  the  divisor;  and  then,  if  the 
two  quantities  have  the  same  sign,  prefix  to  the  quo- 
tient thus  obtained  the  sign  +^  if  different,  the  sign  -. 


I 


Thus  14a5-f-2a=-r— =75,-12a-f-10c=: — =--=-—  -a-f—2c= -f  -j^. 
2a         '  10c      5c'  2c 


18  DIVISION. 

The  rule  for  the  sign  of  the  quotient  is  the  same  as 
that  given  in  (19),  viz.  tliat  like  signs  produce  +  and 
unlike  — ;  and  is  clearly  derived  from  it,  for  if  4-  ^ 
Qnultijplied  by  —  h  produces  —  a5,  of  course  —  db  di- 
vided hj  +  a  produces  —  5 ;  and  so  in  the  other  cases. 

26.  One  power  of  a  quantity  is  divided  by  another 
by  subtracting  the  index  of  the  latter  from  that  of 
the  former. 

Thus  —  =  c&^-»=  a' ;  for  —  =  — r--=  a^ :  so— ^^ —  =  x-yz^, 
a^  a*        a*  xyz 

27.  If  the  dividend  contain  several  terms,  while  the 
divisor  still  consists  of  only  one,  each  term  of  the 
former  must  be  separately  divided  by  the  latter. 

2>xy  oxy       Zxy       Zxy       3  x' 

p     -    a^c^  -  2al)c^  +  3ac'  _      a^c^  ^  2abc^  __  oac^         a      1    2e 

28.  But  if  the  divisor  be  also  a  compound  quantity, 
we  must  proceed  as  in  common  Arithmetic  :  viz. 

(i)  Place  the  quantities,  as  in  Division  of  Arithmetic, 
arranging  the  terms  of  each  of  them,  so  that  the  dif- 
ferent powers  of  some  one  letter,  common  to  both  of 
them,  may  follow  in  order  of  the  magnitude  of  their 
indices,  (it  matters  not  whether  in  ascending  or  de- 
scending  order,  only  the  same  order  in  each  of  them) ; 

(ii)  Divide  the  first  term  of  the  dividend  by  that  of 
the  divisor,  and  set  the  result  in  the  quotient ; 

(iii)  Multiply  the  whole  divisor  by  thefirst  term  of  the 
quotient,  and  subtract  the  product  from  the  dividend ; 

(iv)  Bring  down  fresh  terms  (as  may  be  required) 
from  the  dividend,  and  repeat  the  whole  operation. 


DIVISION.  19 

Ex.  1.  Ex.  2. 

1-   X  ^x^-   Wy 

-12a;y'^  +  16y» 

Ex.  3.  a-x)  a^-x^  {a^  +  ax  +  x^    Ex.  4.  a  +  ic)  a'  +  x^  (a^-<ix  +  x^ 
a^  -  a^x  a^  +  a' a; 


a^x- 

-x^ 

a^X' 

-ax^ 

ax^ 

-x' 

ax' 

-x' 

■  a^x+x" 

■  a^x^ax^ 


ax*-^x' 
ax'^+x* 


Ex.  6.  a+x)  a'^+x^  (a-x+ Ex.  6.  Or-x)  a*  +  jc'Ca  +  x  +  — 

^      2  a  +  x  '  ^  Or^ 

a^+ax  a^  -  ax 


-  ax  +     x^  ax  +    x^ 

-ax-    x^  ax  -    x^ 


2x^  2x^ 

Ir  each  of  the  last  two  examples  we  have  a  remainder  2aj*, 
which  we  place  in  the  quotient,  as  in  Arithmetic,  over  the  divisor, 
in  the  form  of  a  fraction,  thus  indicating  that  2x'^  remains  still  to 
be  divided  hy  a  +  x^  a-x  respectively. 

In  this  and  other  cases,  as  in  common  Arithmetic,  this  fraction 
could  not  be  avoided,  since  a'+x"^  is  not  exactly  divisible  by  a+a; ; 
but  the  student  should  be  cautioned,  that,  unless  attention  is  paid 
to  the  arrangement  according  to  powers^  alluded  to  above  (28  i), 
and  that,  not  only  with  the  dividend  and  divisor  at  starting,  but 
also  throughout  the  sum.  care  being  taken  in  all  Ihe  remainders 
to  preserve  the  order  of  the  indices  of  the  principal  letter,  or  Ze^^er 
of  reference^  as  it  is  called,  there  will  always  be  a  fractional  term 
of  this  kind,  instead  of  a  clear  and  complete  quotient. 
Ex.  12. 

1.  Divide  al'c^  by  alc^  -  KS^x'^y'"  by  -  oZxy^^  -lOalx^y  by  2ax'y. 

2.  Divide  6x^y  -  4x^2  +  6xyz  by  2a;,   Sa'S' -  35a^5V  +  20aJc*  by 
-  5a5,  and  a^x'^y  -  Za'^bx^y  +  Za¥xy'  -  ¥xy^  by  ahxy. 


I 


20  DIVISION. 

8.  Divide  2m^n^  -Zm?i^  +  im^n-n*  by  ^Zmn^,  and 
-  Za'h  +  5a'h'  -  Ga'h'  -  aV  +  Al'  by  -  2a''l\ 

4.  Divide  a;^  +  6a;  +  5  by  aj  +  1,  and  m'  -  Gtti'*  +  11m  -  G  by  in  -  2. 

5.  Divide  G^'  -1G«5  +  8&'  by  2a-4&,  and  tr''  +  ISxy  +  Gy*   by 

2x  +  Zy,  G.  Divide  ^a'^l''  -  a¥  -  125*  by  2>ab  +  W. 

7.  Divide  aUW  by  «-  +  2a6+22*',  and  4a?V  +  l  by  2j; V-2a;?/ + 1. 

8.  Divide  a;''-2a;V+2xV'-4a;V'  +  8a;'^2/'  +  lG2jy'-32y*  by  a;^-2y^ 
0.  Divide  l  +  Q^x^  +  hx^  by  l  +  2a;+a;^  and  ^^"-6^  +  5  by  a'-2«+l. 

1 0.  Divide  a;'*-4a;2/^  +  Zy^  by  x^-2xy + y'.  and  m*  +  4/71 + 3  by  m^ + 2m  + 1. 

11.  Divide  aJ'-Aa^l'' -Wl''-\lab^-\2l''  by  a^-2a&-32;^ 

12.  Divide  a;«-2a;Vl  by  a;--2a;+l,  and  «°  +  2a'6^  +  5»  by  aV2a5  +  Z''. 


29.  In  some  of  the  following  Examples,  the  div'  and  div*  are  not 
properly  arranged  according  to  powers :  the  student  must  attend 
to  this  before  and  i)i  the  course  o/ division.  In  Ex.  1,  for  instance, 
where  a  is  taken  as  the  letter  of  reference,  and  its  powers  arranged 
in  descending  order,  there  is  found  in  the  first  rem'  the  terms 
-  a"^!)^  -  crc.  These  terms  must  be  set  Jirst,  but  since  both  involve 
a^,  there  is  nothing  as  far  as  a  is  concerned  to  shew  which  is  to  be 
set  first  of  the  two.  In  such  cases  we  take  another  letter,  as  5,  to 
be,  as  it  were,  next  in  authority  to  a,  and  so,  (arranging  in  de- 
scending powers  of  &,)  we  prefer  -  a^l)  to  -  a'c. 

Ex.  1.  Divide  a^  +  l^  +  c^  -Zahc  by  a  +  h  +  c' 

a  +  h  +  c)  a^  -  Zabc  +  5^  +  c^  {ci^  -  ah  -  ac  +  1^-1)0  +  c" 
«'  +  (c^b  +  a^c 

-  cc'b  -  a^c  -  oabc 

-  a^b  -  ab'^  -    cibc 

-  o^c  +  a¥  -  2alc 

-  a'c  -  abc  -    ac^ 

+  al)^  -  abc  +  ac^  +  5* 
+  a?>'  +    &'  +  b'^c 

-  abc  +  ac^  -  b'^c 

-  abc  -  b^c  -  bc^ 

+  ac"^  +  bc^  +  c' 
+  flrc^  +  be'  +  c' 

The  above  is  the  most  easy  method  in  such  a  case ;  but  thv 
following,  in  which  the  cocff  •  of  the  difterent  powers  of  a  are  col 
Iccted  in  brackets,  is  the  most  neat  and  compendious. 


DIVISION.  21 

Ex.  2.  a  +  (h  +  c)]  a*  -  Sahc  +  (h*  +  c')  [a'  -  (6  +  c)  a^  +  (6^  -  Jc  +  c') 
«'  +  (&  +  c)  fl^' 

-  (6  +  c)  flt'*  -  3&C6^ 

-(b  +  c)  a^  -  (6"  +  2bc  +  c^)  <^ 

+  (6^  -  5c  +  c^)  a  +  (6'  +  c*) 

Ex.  13. 

1.  Divide  a;' -(a  +p)x^  +  (q  +  ap)  x - aq  hy  x- a, 

2.  Divide  maz^  +  (mb  -  no)  z^  -  (mc  +  nb)  z  +  nchy  mz-  n, 

3.  Divide  y^  -  my*'  +  ny^  -  ny^  +  wzy  -Ihj  y  -1. 

4.  Divide  a^ - 6^  - c'  +  ^' - 2acZ  +  2bc  hy  a-b  +  c-d. 

5.  Divide  a^  +  ab  +  2ac- 26'  +  75c -Zc^hy  a-b  +  3c. 

6.  Divide  a^-6^+c'  +  3a6c  by  a-b+c.  and  a'-^'-c^-SaSc  by  Or-b-e, 

7.  Divide  i + x^Sy^+ 6xy  by  1 + ic-2y,  and  1-x^ + 8y '  +  6xy byl-a; + 2y. 

8.  Divide  a;'  -  By'  -  27^^  -  ISxyz  hyx-2y-  3z. 

9.  Divide  x*  +  y^-z*  +  2xY -2z^-l  by  x^  +  y''-z''- 1. 

10.  Divide  «  by  1  +  a;,  and  1  +  2ic  by  1  -  3a?,  each  to  4  terms  in 

the  quotient. 

11.  Divide  1  by  1  -  2aj  +  a;',  and  1  -  aa;  by  1  +  bx,  each  to  4  terms. 

12.  Find  the  rem',  when  x^  -px^  +  qx-r  is  divided  by  aj  -  a. 

30.  We  have  seen  above  (28  Ex.  3  and  4)  that  a^-x^ 
is  exactly  divisible  by  a-x,  and  a^'  +  x^  hj  a  +  x,  and 
that,  in  the  quotient  in  each  case,  the  powers  of  a 
decrease  continually,  while  those  of  x  increase.  The 
following  general  facts  should  be  well  noticed,  as  they 
will  enable  us  to  write  down  at  once  the  quotient,  when 
similar  cases  occur,  as  they  often  do,  in  practice.  It 
will  be  seen  that  the  index  n  is  here  used  to  denote 
generally  any  index,  as  the  case  may  be :  the  quantity 
a^  is  called  the  n^^  power  of  a,  and  read  a  to  the  n^\ 
If  the  index  be  odd^  a^+cc"*  (like  a-]-x)is  div.  by  a+x^ 

gn __  ^n ^]i]^(3  a-x)   .  .  .  .  by  a  -  aj ; 
iftheindexbe^'y^/i, (2^+a?'*(likea''+a?'')  .  ,  .  .hj neither^ 

a'»-a;~(like^'~a?') by  loth. 

The  student  will  best  remember  these,  by  thinking,  in  each 
case,  of  the  simplest  form  of  the  same  kind. 


ft 


22  DIVISION. 

Thus,  for  a^  +  x^  (index  even.,  sign  +)  let  him  think  of  a*  +  x'; 
this,  he  knows,  is  div.  by  neither  ;  then  a*  +  a;*  is  div.  by  neither : 
again,  for  a'  -  x^  (index  odd^  sign  -)  let  him  think  of  <x  -  a; ;  this 
is,  of  course,  divisible  hy  a  -x\  then  a*  -x^  is  so  divisible:  for 
a*  -  a;",  let  him  think  of  a^  -x'\  this  is  divisible  by  hoth  a  +  t^ 
and  a-x:  then  a^ - x^  is  divisible  by  both. 

Now,  in  every  case,  tlie  quotient  will  consist  (as  may 

be  seen  by  actual  division)  of  terms,  in  which  the 

powers  of  a  decrease,  and  of  x  increase  continually  : 

but  when  the  div*"  is  a-x,  these  terms  are  all  +; 

when  it  is  ^  -f-  Xy  they  are  alternately  +  and  -. 

^,       a*-x*       ,       .  „       ,  a*-a;*        ,       „  ,       . 

Thus =  a'  +  a^x  +  ax"  +  x^, =  a'  -  a'x  +  ax^  -  x*, 

a-x  a  +  X 

a*  -  OjX  +  arx^  -  ax^  +  x*, 

a  +  X 

The  above  results  may  now  be  applied  to  muny 
similar  cases. 

Ex.  1.  ^'?^^=4aV  +  2«a'-f  1.        Ex.  2.  ^J^'  =a;^-3a^+ V- 

Ex.  14. 
Write  down  the  quotients 

1.  Of  a^  -X  ^  by  (5^  +  x^  a^  -  x^  by  a  -  ir,  and  «"  -  a;"  by  «a^  +  x. 

2.  Of  9a;'  - 1  by  3a;  -  1,  25a;'  -  1  by  5a;+l,  and  4a;'  -  0  by  2a;+3. 

3.  Of  9w'7i'  -  25  by  Zmii  +  5,  and  16m*  -n*  by  4m'  +  7j,'. 

4.  Of  l  +  8a;^  by  l+2a;,  27a;^-l  by  3a;-l,  and  l-16a;*  by  l  +  2a;. 

5.  Of  a;*-81y*  by  a;-32/,  ^'^+326'  by  a+2h,  and  a;"-y»-'  by  a;»+y'. 

6.  Of  ^ct'  +  l^  by  -^-a  +  &,  and  a;*i/*  -  s*  by  xy  +  2. 

7.  Of  {a  +  hy  -  c'V  a  +  h  -  c,  nud  a^  -  (b  -  cy  hj  a  -h  -i-  c. 

8.  Of  (a;  +  2/)^  +  z^hy  X  +  y  +  z,  and  x^  -  (y  -  zy  by  x-y+z. 


31.  The  above  results  and  those  of  (23)  may  also 
be  applied  to  resolve  algebraical  quantities  into  their 
elementary  factors,  a  j^rocess  which  is  often  required. 

Ex.  1.  4a;'  -  y'  =  (2a;  +  y)  (2x  -y):  a;'  +  8  =  (a;  +  2)  (a;'  -  2a;  +  4). 
Ex.  2.  (2a-hy-(a-2by=(2a-h-^a-2b)  (2a-6-a  +  25)  =  3  (a^b)  (a  +  b), 
Ex.  3.  a;"-«*=(a;'+a')  (x^-a^)=(x+a)  (x^-ax+x"^)  (x-a)  (x^  +  ax-i-x^). 
Ex.  4.  (a'  -  xy  =  \(a-x)  (a'  +  ax  +  a;') }'--=(« -at)' x (a'  +  aa;  +  a;')'. 


DIVISION.  2o 

Ex.  16. 

Resolve  into  elementary  factors 
1.1-  4:x\  a"  -  ^x\  9m^  -  An\  2baH''  ~  4a;^  l^xY  -  25xY' 

2.  x^  +  y%  a;'  -  y',  1  +  ic'i/',  a;*  -  1,  a^xy^  -  x^y,  la^Vc  -  Sah^cK 

3.  25x^  -  aV,  a«  -  9a^5«,  8x»  -  27,  a»  -  85',  a'icV  +  27a;'y*. 

4.  oj*  +  32,  a'ic'  +  27a;«,  8a;'  +  y«,  a*6"  -  c\  a^bc  +  2o^'5c'  +  adc\ 

5.  81a;*  -  1,  a;«  -  64,  x*  -  2hx'  +  5  V,  a;«  -  2a'a;*  +  a'x\ 

6.  (3a;  -  2)*  ~  (a;  -  3)^  («  +  5)^  -  Ah\  (4x  +  3y)''  -  (3a;  +  4y)\ 

7.  (x^  +  y'y  -  AxHj\  c^-(a-  ly,  (2a  +  hy  -  (2a  -  5)^ 

8.  x^  +  y^  +  Zxy  (x  +  2/),  ^'  -n^  -m  (m^  -n^)  +  n(m-  7i)\ 

a^-ab  +  2  (5^  -  «2>)  +  3  («'-  5')  -  4  (a-hy. 

9.  5(a!'-t/=)H  3(a;+t/)^  3  (a;^  -  y^  -  5  (a;  -  ^)^ 

(x  +  yy  +  2  (x^  +  xy)  -  3  (a;'  -  y^), 
10.  2  («'  +  a'h  +  a&^)  -  (a'-¥),  a'  -h'-  Zal  (a  -  l\ 
a'-V  +  (a''  -  ly  -  2a'  +  2a''l\ 

32.  So  too  we  may  often  apply  (23  iii)  to  resolve  a 
trinomial  into  factors. 

Ex,  1.  a;H7a;+12=(a;+3)  (a;+4).    Ex.  2.  a;^-9a;+14=(a;-2)  (a;-7). 
Ex.  3.  flj^_5a;-14=(a;-7)  (a;+2).    Ex.  4.  6a;^+a>-12=(3a;-4)  (2a;^3). 

The  student  may  notice  that,  if  the  last  term  of  the  given 
trinomial  be  positive^  (Ex.  1,  2),  then  the  last  terms  of  the  two  fac- 
tors will  have  the  same  sign  as  the  middle  term  of  the  trinomial ; 
but  if  negative^  (Ex.  3,  4),  they  will  have  one  the  sign  +,  the 
other  -. 

In  Ex.  4,  it  is  clear  that  the  first  terms  of  the  two  factors 
might  be  (Sx  and  a;,  or  3a;  and  2x,  since  the  product  of  either  of 
these  pairs  is  Qtx"^ ;  and  so  the  last  two  terms  might  be  12  and  1, 
6  and  2,  or  4  and  3 :  it  is  easily  seen  on  trial  which  are  to  be 
taken,  that  is.  which  serve  also  to  produce  the  middle  term  of  the 
trinomial. 

Ex.  16. 

Resolve  into  elementary  factors 

1.  a;H6a;+5,  a;H9a;+20,  a;''-5a;+6,  a;^-8aj+15,  a;^  +  8a^+7,  a;'-10;r+9. 

2.  a;^  +  X-Q,  x^'^x^^,  a;^-2a;-3,  a;''+2a;-15,  a;V7a;-8,  a;^-8a;-9. 

3.  4a;'»  +  8a;+3,4a;'*  +  13a;+3,4a;^  +  lla;-3,4a;'~4a'-3, 3a;^+4a!-4  6a!V5a;-4. 

4.  12a;^-5a;-2,  12a;' -  14a;  +  2,  12a;'-a;-l,  a;' +  a;-12,  2r'-2a;-5. 

5.  aV-3a'a;+2a*,  a'-«'^+6aa;^3a'5+a'5'-2«5^  12a*  +  rzV-a;*. 

6.  2a;V+5a;'y'+2a;y',9a;V'-3a;y'-6y*,  6aV+a5a;-a',  65V-75a;'^a;*. 


I 


OHAPTEE   III. 

I' 

SIMPLE   EQUATIONS. 

33.  When  two  algebraical  quantities  are  connected 
by  the  sign  =,  the  whole  expression  is  called,  accord- 
ing to  circumstances,  an  identity  or  an  equation. 

An  identity  is  merely  the  statement  of  the  equiva- 
lence of  two  diflferent  forms  of  the  same  quantity,  and  is 
true  for  any  values  of  any  of  the  letters  involved  in  it. 

Thus  it  is  always  true,  whatever  be  the  values  of  x  and  y,  that 
(x  +  y)  (x-  y)=x'^-  2/^5  or  that  (x  ±  yY  =  aj^  ±  2xy  +  y"^ :  and  so  also 
it  is  always  true  that  \  (x  +  y)  +i(:x-y)  =  ^  +  ^y  +  ^x-^y  =  Xj 
and,  in  like  manner,  that  ^  ix+y)-^{x- y)  =^x+^y-^x+iy=y  : 
each  of  these  expressions  is  therefore  an  identity. 

And  in  this  way  we  may  see  one  of  the  principal  ad- 
vantages of  Algebra,  viz.  that  it  enables  us  to  prove 
once  for  all,  and  express  by  means  of  letters  as  general 
statements,  results  which  by  mere  Arithmetic  we  could 
only  shew  to  be  true  upon  actual  trial  in  each  instance. 

Of  this  we  have  seen  examples  in  the  three  formulas  of  (23)  ; 
and  so  also  the  two  last  above  given  express  the  general  facts, 
that  the  greater  (x)  of  any  two  quantities  is  equal  to  the  surn^  and 
the  lesser  (y)  is  equal  to  the  difference,  of  their  semi-sum  and 
semi-difference. 

34.  An  equation^  however,  is  the  statement  of  the 
equality  of  two  differe^it  algebraical  quantities ;  in 
which  case  the  equality  does  not  exist  for  any^  but 
only  for  some  particular  values  of  one  or  more  of  the 
letters  contained  in  it. 

Thus  the  equation,  a;  -  3  =  4,  will  be  found  true  only  when  we 
give  X  the  value  7,  and  x^  =  Sx  -  2  only  when  we  give  x  the  value 
1  or  2. 


SIMPLE    EQUATIONS.  25 

We  arc  about  to  explain  the  method  of  finding  these 
values  which  satisfy  the  simpler  kinds  of  equations. 

35.  The  last  letters  of  the  alphabet  a?,  y,  2^  (fee,  are 
usually  employed  to  denote  those  quantities,  to  which 
particular  values  are  to  be  given  in  order  to  satisfy  the 
equation,  and  are  said  to  be  the  unJcnown  quantities. 

An  equation  is  said  to  be  satisfied  by  any  value  of 
the  unknown  quantity  which  makes  the  values  of  the 
two  sides  of  the  equation  the  same. 

This  includes  the  case  when  all  terms  of  an  equation  lie  on  one 
side  and  0  on  the  other,  as  in  a;^  -  3ic  +  2  =  0,  which  is  satisfied  by 
1  or  2j  cither  of  which,  being  put  for  x^  makes  the  first  side  =  0. 

Those  values  of  the  unknown  quantities,  by  which 
the  equation  is  satisfied,  are  called  its  'roots. 

Thus  1  and  2  are  the  roots  of  the  equation  aj^  -  3a;  +  2  =  0,  7  is 
the  root  of  a;-3=4,  1,  2,  and  3  are  the  roots  of  a;'-6=6aj^-lla;j  &c. 

36.  An  equation  of  one  unknown  quantity  is  said 
to  be  of  as  many  dimensions  as  is  denoted  by  the  index 
of  the  highest  power  of  the  unknown  involved  in  it. 

Thus  aj  -  3  =  4  is  an  equation  of  one  dimension,  or  a  simph 
equation ;  a;'  =  3a;  -  2  is  of  two  dimensions,  or  a  quadratic  equa- 
tion ]  x^~6  =  6x^  is  of  three  dimensions,  or  a  cubic  equation ;  x*-  4aj 
=13  is  of  four  dimensions,  or  a  Mquadratic  equation;  &c.  &c. 

It  may  be  noticed,  in  passing,  that  it  can  be  proved 
tliat  every  equation  of  one  unknown  quantity  has  as 
many  roots  as  it  has  dimensions,  and  no  more. 

37.  Every  term  of  each  side  of  an  equation  may  he 
multiiMed  or  divided  hy  the  same  quantity^  without 
destroying  the  equality  expressed  hy  it. 

Thus,  if  3a;  +  |a;  =  34,  multiplying  every  term  by  4,  we  have 
12a;  +  5a;  =  13G,  or  17a;  =  13G ; 
therefore  also,  dividing  each  term  by  17,  x  =  ^-^  =8. 

I 


26  SIMPLE   EQUATIONS. 

Again,  if  12ir  +  6a;  =  144,  dividing  every  term  by  6, 
2x^  aj  =  24,  or  3a;  =  24; 
hence  also,  dividing  each  term  by  3,  x  =  8. 
We  find,  therefore,  that  8  is  the  root  of  each  of  these  two  equations. 

38.  Hence  an  equation  may  be  cleared  of  fractions^ 
by  multiplying  every  term  by  any  common  multiple 
of  all  the  den".  If  the  l.  c.  m.  be  employed,  the  equa- 
tion will  be  expressed  in  most  simple  terms. 

Thus,  \^  \x  +  \x  +  |a;  =  13,  multiplying  every  term  by  12,  which 
is  the  L.  c.  M.  of  2,  3,  4,  we  have 

¥^  +  V^^  +  V^^  =  1^^>  ^^  (jx  +  Ax  -^Zx-  150 ; 
hence  13a;  =  15G,  and  x  =  V¥  =  12. 

39.  A  qumitity  onay  he  transferred  from  one  side  of 
an  equation  to  the  other  hy  changing  its  sign^  without 
destroying  the  egiiality  exjyressed  hy  it. 

Thus  ii^x — a=^y-{-l)^  adding  a  to  each  side  of  the  equa- 
tion (which,  of  course,  will  not  destroy  the  equality)  we 
have  x—y+l)+a^  and,  subtracting  J  from  each  side, 
we  have  x—h^-y+a  ;  where  w^e  see  that  the  — a  has 
been  transferred  to  the  other  side  with  its  sign  changed 
to  +,  and  so  also  the  +5,  w^tli  its  sign  changed  to  — . 

Hence  if  the  signs  of  all  the  terms  of  both  sides  of 
an  equation  be  changed,  the  equality  expressed  by  it 
will  not  be  destroyed. 

Simple  Eguations  of  one  unhnown  Quantity. 

40.  To  solve  a  simple  equation  of  one  unknown. 

(1)  Clear  it,  when  necessary,  of  fractions  (38) ; 

(2)  Collect  all  the  terms  involving  the  unknown 
quantity  on  one  side  of  the  equation,  and  the  known 
quantities  on  tlie  other,  transposing  them,  when  ne- 
cessary, wirti  change  of  sign  (39) ; 


SIMPLE    EQUATIONS.  37 

(3)  Add  together  the  terms  of  each  side,  and  divide 
the  sum  of  the  known  quantities  by  the  sum  of  the 
coefScients  of  the  unknown  quantity;  and  thus  the 
root  required  will  be  found. 

Ex.  1.  4x  +  2  =  Zx  +  4. 
There  being  no  fractions  here,  we  have  only  to  collect  the  terms ; 
.*.  4x  -  3ic  =  4  -  2j  or  a;  =  2,  the  root  of  the  equation. 

Ex.  2.  4jj  +  5  =  10.C  -  16. 
Here  lOaj  -  4.^  =  5  +  16  ;  .-.  6x  =  21,  and  aj  =  ^'-  =  3f  =  31 

Ex.  3.  5  (a?  +  1)  -  2  =  3  (a;  -  5). 
Here,  removing  the  brackets,  5aj  +  5  -  2  =  3a;  -  15  ; 
.-.  collecting  terms,  5a;-3a;=- 15-5  +  2,  or  2a;=-18,  and  .*.  x=-9. 

Ex.  4.  hx-r2x-a  =  Sx  +  2c. 

Here  2>a;  +  2a;  -  3a;  =  (5  -  1)  a;  =  a  +  2c ;  ,\  x=  -^ — ::-. 

Ex.  17. 

1.  4a;-2  =  3a;  +  3.        2.  3a; -f  r  =  9a;-5.        3.  4a;  +  9  =  8a;-3. 
4.  3  +  2a;  =  7  -  5a;.        5.  a;  =  7  +  15a;.  6.  mx  +  a  =  nx  +  d. 

7.  3(a;-2) +  4  =  4(3-a;).  8.  5-3(4-a;)  +  4(3-2a;)  =  0. 

9.  13a;  -21  (a'-3)  =10-21  (3  -  a;).     10.  5  (a  +  x)-2x=  ^  (a-  5a;). 

11.  3(a;-3)-2(a;-2)  +  a;-l  =  a;  +  3  +  2  (a;  +  2)  +  3(a;  +  1). 

12.  2a;-l-2(3x-2)  +  3(4a; - 3) - 4(5a; - 4)  =  0. 

13.  (2  +  a;)  (a  -  3)  =  -  4  -  2aa;.        14.  (m  +  n)  (m  -x)  =  m  (n  -  x). 


I 


Ex.  5.  ia;-fa;  +  fa;  =  11  +  |a;. 

Here  we  first  clear  the  equation  of  fractions,  by  multiplying 
every  term  by  24,  the  l.  c.  m.  of  the  den",  and  (observing  that  in 
the  first  fraction-^/-  =  12,  in  the  second,  -^^  =  8,  and  so  in]  the 
others)  thus  we  get  12a;  -8x2a;+6x8a;  =  264  +  3a;,  or  12a;-16a5 
+  18a;  =  264  +  3a;  collecting  terms, 

12a;-  16a;  +  18a;  -  3a;  =  264 ;  .-.  11a;  =  264,  and  x==-^^  =  24. 

Ex.  6.  l(a;  +  1)  +  »  (a;  +  2)  =  16-^  (.r  +  3). 

Multiplying  by  12,  we  have 
6  (a;  +  1)  +  4  (a;  +  2)  =  192  -  3(a;+3),   or  Ox  +  6+4a;+8=192-3x-9; 
collecting, 

6a;  -f  4a;  +  3a;  =  192-9-  6  -  8  ;  .-.  13a;  =  169,  and  x  ^  Vr  =  1^. 


28  SIMPLE  EQUATIONS. 

Ex.7. 

1  (Zx'+xy^  i2xUx)  +  l  (x'^xy2^\=x'^-f^+i  ix'-^x)-^(x'^5x). 
Expressing  the  mixed  number  2/^  as  an  improper  fraction  f  J, 
we  then  multiply  by  GO,  the  l.  c.  m.  of  the  den";  and,  olserting 
the  remarh  at  the  end  of  (l(j)^  we  thus  obtain 

90ic'  +  30iB-40a;''-20iC + 15a;^  +  15x-129=G0a;'  +  8  +  10a;'  +  10aj-5a;'-25aj ; 

collecting,  we  find  that  the  terms  involving  x^  destroy  one  another 
(otherwise  the  equation  would  be  a  quadratic),  and  we  have  the 
result 

30a;  -  20a;  +  15a;  -  10a;  +  25a;  =  129  +  8 ;  /.  40a;  =  137,  and  x  =  3^  J. 
Ex.  18. 
1.  la;+^a;=a!-7.        2.  •|a;-^a;={aj~l.        3.  Aa;— Ja;+|a;=2-Ja;+/^ic• 
4.  |a;  +  ^(a;-2)  =  2a;-7.  5.  |a;  +  i(a;-l)  =  a;-4.  ^ 
6.  i(9-2a;)  =  ^-yV(7^-18).         7.  a;  +  i  (14  -  a;)  =  ^  (21  -  a;). 
8.  2a;~^=f(3-2a;)+|a;.        9.  |  (2a;  +  7)-yV  (9«^-8)=H«-ll)• 
10.  \{x^a)-\C2x'-Zl)-\{a-x)  =  Q. 
11.  i(3a;-l)-f(a;-l)  =  H^-3)-^(a;-5)  +  5i 
12.ia;-lf=8?  +  2(?a;-l)-^(a?  +  8). 


In  some  of  the  following  examples  the  common  multiple  of  all 
the  den"  is  too  large  to  be  conveniently  employed.  In  such  a 
case,  we  may  see  whether  two  or  three  of  the  den"  have  a  simple 
common  multiple,  and  get  rid  of  their  fractions  first,  observing  to 
collect  terms,  and  simplify  as  much  as  possible,  after  each  step. 

Ex.  8.  tV  (2a;  +  3)-^(a;-12)  +  J  (3a;  +  1)  =  5J-  +  yV  (4^  +  3). 
Here  the  l.  c.  m.  of  all  the  den"  would  be  132 :  but  as  12  will 
include  three  of  them,  multiplying  by  it,  (having  first  changed 
H  to  ^{^,  we  get 

If  (2a;  +  3)  -4  (a;-12)  +  3  (3a;  +  1)  =  64  +  4a;  +  3; 

/.  If  (2a;  +  3)-4a;  +  48  +  9a;  +  3  =  64  +  4a;  +  3; 

hence,  collecting  terms  and  simplifying,  we  have 

ff  (2a;  +  3)  -4a;  +  9a; -4a;  =  64  +  3-48-3,  or  jf  (2a;  +  3)  +  a;=16,- 

.-.  12  (2a;  +  3)  +  11a;  =  176,  or  24a;  +  11a;  =  176  -  36 ; 

.-.  35a;  -  140,  and  x  =  W  =  4. 


I 


SIMPLE   EQUATIONS.  29 

Ex.  19. 

1.  A(2a5-3)-J(3.T-2)  =  |(4aj-3)-3/^. 

2.  5(^_9)^7(^.5)  =1(05-7)  + If. 

3.  tV(2^-1)-iV(3^-2)  =  tV(^-12)-3V(^  +  12). 

4.  I  (7x  +  20)  -j\  (3aj  +  4)  =  yV  i^^  +  1)  -  ^V  (20  -  8a;). 

5.  f  (^-2x)-f  (2«-a')  +  ^(a;-a)  =  J-|(«+ir). 

6.  ,V(9^-10)-TV(2^-7)  =  f^-3V(5  +  aO. 

7.  A(4^-l)-A(2^+  l)  =  51-/2^- 

8.  |{a-(5-a;)}-J|oj-(5-a)}-f{5-(^+aj)}=||a;  +  a-5}. 

9.  |(4aj-21)  +  7|  +  |(a;-4)  =  a;  +  3f-i(9-7a;)  +  7V- 

10.  -J  (a;  -  a)  -  -5^^  {m  -  (a  -  aj) }  =  ^  (m  +  ic)  -  ^V  (^^^  +  lOiii), 

11.  J^  (2a;  +  7)-yV  (S.'c-T)  =  1|- 2V  (3aJ  +  4). 

12.  i \x-'(2a-Zc)\  -i^-^{7a-5  (x-2c)\  =  ^\{S (a+lOc) '-(2c-x)\. 

Problems  jproducing  Simjple  Eqitations. 
41.  We  shall  now  see  the  practical  application  of  the 
above  in  the  solution  of  many  entertaining  Arithmeti- 
cal questions.  In  treating  these,  however,  ^ter  having 
observed  the  methods  used  in  the  following  examples, 
the  student  must  be  left  very  much  to  his  own  inge- 
nuity, as  no  general  rule  can  be  stated  for  their  solu- 
tion. The  only  advice  that  can  be  given  is  to  read  over 
carefully  and  consider  well  the  meaning  of  the  ques- 
tion proposed ;  then  it  will  always  appear  that  som.e 
quantity,  at  present  unknown,  is  required  to  be  found 
from  the  data  furnished  by  it :  put  x  to  represent  this 
quantity,  and  now  set  down  in  algebraical  language  the 
statements  made  in  the  question,  using  x  whenever 
this  unknown  quantity  is  wanted  in  it.  AVe  shall 
thus  (in  the  problems  w^e  are  now  considering)  arrive 
at  a  simple  equation,  by  means  of  which  the  value  of 
X  may  be  found. 

Ex.  1.  What  number  is  that  to  which  if  8  be  added,  one-fourth 
of  the  sum  is  equal  to  29  ? 
Let  X  represent  the  number  required  ; 


30  SIMPLE  EQUATIONS. 

adding  8  to  it,  we  have  a;  +  8,  one-fourth  of  this  is  J  (aj  +  8),  and 

this  is  equal  to  29  ; 
we  have,  therefore,  the  equation  ^  (x  +  S)  =  29,  whence  x  =  108. 

Ex.  2.  What  number  is  that,  the  double  of  which  exceeds  its 
half  by  6? 

Let  X  =  the  number ; 

then  the  double  of  x  is  2x,  the  half  of  a;  is  ^a* ; 
hence  2x-\x  =  6,  whence  x  =  A. 
Ex.  3.  A  cask,  which  held  270  gallons,  was  filled  with  a  mix- 
ture of  brandy,  wine,  and  water.     There  were  30  gallons  of  wine 
in  it  more  than  of  brandy,  and  30  of  water  more  than  there  were 
of  wine  and  brandy  together.     How  many  were  there  of  each  1 
Let         X  -  no.  of  gals,  of  brandy ; 
.*.  a;  +  30  =     •     .     .     .    wine, 
and  2a;  +  30  =     ....    wine  and  brandy  together ; 
.*.  2a;  +  30  +  30  or  2a;  +  GO  =  gals,  of  water  5 
but  the  whole  number  of  gallons  was  270 ; 

.-.  a;  +  (a;  +  30)  +  (2a;  +  60)  =  270, 
wlience  x  =    45,  the  no.  of  gals,  of  brandy, 

a?  +  30  =    75, wine, 

2a;  +  CO  =  150, water. 

Ex.  4.  A  sum  of  £50  is  to  be  divided  among  A^  B,  and  C,  so 
that  A  may  have  13  guineas  more  than  i?,  and  C  £5  more  than 
A  :  determine  their  shares. 
Let  X  =  jS's  share  in  sMllings  : 

.-.  a;  +  273  =  ^'s,  and  {x  +  273)  +  100  or  a;  +  373  =  C's ; 
.-. ,  since  £50  =  IOOO5,  {x  +  273)  +  x  +  {x-^  373)  =  3a;  +  646  =  1000 ; 

.-.  3a;  =  354,  and  x  =  118,  a;  +  273  =  391,  x  +  373  =  491, 
and  the  shares  are  391«,  118^,  491^,  or  £19  II5,  £5  18s,  £24  11«, 
respectively. 

Ex.  5.  A^  B,    C  divide  among  themselves  620  cartridges,  A 
taking  4  to  j5's  3,  and  6  to  6"s  5  :  how  many  did  each  take  ?  ^ 
Let  X  -  A^s  share  ;  then  fa;  =  B^s,  |a;  =  6"s ; 

.-.  x  +  ^x  +  ix=^  620,  whence  x  =  240,  J;r  =  180,  ix  =  200. 
AYe  might  have  avoided  fractions  by  assuming  12a;  for  A's  share, 
when  we  should  have  had  9a;  =  i?'s,  and  10a;  -  C^s; 
.-.  12a;  +  9a;  +  10a;  =  620,  whence  a;  =^  20  ; 
and  the  shares  are  240,  180,  200,  as  before. 


SIMPLE   EQUATIONS.  31 

Ex.  20. 

1.  What  number  is  that  which  exceeds  its  sixth  part  by  10  ? 

2.  What  number  is  that,  to  which  if  7  be  added,  twice  the  sum 
will  be  equal  to  32? 

3.  Find  a  number  such  that  its  half,  third,  and  fourth  parts 
shall  be  together  greater  than  its  fifth  part  by  106. 

4.  A  bookseller  sold  10  books  at  a  certain  price,  and  afterwards 
15  more  at  the  same  rate,  and  at  the  latter  time  received  35s. 
more  than  at  the  former :  what  was  the  price  per  book  ? 

5.  What  two  n^s  are  those,  whose  sum  is  48  and  difference  22  ? 

6.  At  an  election  where  979  votes  were  given,  the  successful 
candidate  had  a  majority  of  47 ;  what  were  the  numbers  for  each  ? 

7.  A  spent  2$  6d  in  oranges,  and  says,  that  3  of  them  cost  as 
much  under  Is,  as  9  of  them  cost  over  1^:  how  many  did  he  buy  ? 

8.  The  sum  of  the  ages  of  two  brothers  is  49,  and  one  of  them 
is  13  years  older  than  the  other :  find  their  ages. 

9.  Find  a  number  such  that  if  increased  by  10,  it  will  become 
five  times  as  great  as  the  third  part  of  the  original  number. 

10.  Divide  150  into  two  parts,  so  that  one  of  them  shall  be 
two-thirds  of  the  other. 

11.  A  post  is  a  fourth  of  its  length  in  the  mud,  a  third  of  its 
length  in  the  water,  and  10  feet  above  the  water ;  what  is  its 
length  ? 

12.  There  is  a  number  such  that,  if  8  be  added  to  its  double, 
the  sum  will  be  five  times  its  half.     Find  it. 

13.  Divide  87  into  three  parts,  such  that  the  first  may  exceed 
the  second  by  7,  and  the  third  by  17. 

14.  Find  a  number  such  that,  if  10  be  taken  from  its  double, 
and  20  from  the  double  of  the  remainder,  there  may  be  40  left. 

15.  A  market-woman  being  asked  how  many  eggs  she  had,  re- 
plied, If  I  had  as  many  more,  half  as  many  more,  and  one  egg 
and  a  half,  I  should  have  104  eggs :  how  many  had  she? 

IG.  A  and  B  began  to  play  with  equal  sums  ;  A  won  30*,  and 
then  7  times  A^s  money  was  equal  to  13  times  J5's :  what  had 
each  at  first  ? 

17.  A  is  twice  as  old  as  B;  twenty-two  years  ago  he  was 
three  times  as  old.     Required  J.'s  present  age. 

18.  A  and  B  play  together  for  a  stake  of  5^  ;  if  A  win,  he  will 


$2  SIMf»LE  EQUATIOKS. 

have  thrice  as  much  as  B ;  but  if  he  lose,  he  will  have  only  twice 
as  much.     What  has  each  at  first  ? 

19.  Divide  £G4  among  three  persons,  so  that  the  first  may  have 
three  times  as  much  as  the  second,  and  the  third,  one-third  as 
much  as  the  first  and  second  together. 

20.  A  workman  is  engaged  for  28  days  at  2s  Qd  a  day,  bub 
instead  of  receiving  anything,  is  to  pay  Is  a  day,  on  all  days  upon 
which  he  is  idle :  he  receives  altogether  £2  12s  (jd ;  for  how  many 
idle  days  did  he  pay  ? 

21.  A  person  buys  4  horses,  for  the  second  of  which  he  gives 
£12  more  than  for  the  first,  for  the  third  £G  more  than  for  the 
second,  and  for  the  fourth  £2  more  than  for  the  third.  The  sura 
paid  for  all  was  £230.     How  much  did  each  cost  ? 

22.  A  person  bought  20  yards  of  cloth  for  10  guineas,  for  part 
of  which  he  gave  lis  (xZ  a  yard,  and  for  the  rest  7*  Qd  a  yard. 
IIow  many  yards  of  each  did  he  buy  ? 

23.  Two  coaches  start  at  the  same  time  from  York  and  London, 
a  distance  of  200  miles,  travelling  one  at  9|  miles  an  hour,  the  other 
at  9| :  where  will  they  meet,  and  in  what  time  from  starting  ? 

24.  A  cistern  is  filled  in  20  min.  by  3  pipes,  one  of  which  con- 
veys 10  gallons  more,  and  another  5  gallons  less  than  the  third 
2)er  minute.  The  cistern  holds  820  gallons.  How  much  flows 
through  each  pipe  in  a  minute  ? 

25.  A  starts  upon  a  walk  at  the  rate  of  4  miles  an  hour,  and 
after  15'  B  starts  at  the  rate  of  4J  miles  an  hour ;  when  and 
where  will  he  overtake  A  1 

26.  A  garrison  of  1000  men  was  victualled  for  30  days ;  after 
10  days  it  was  reinforced,  and  then  the  provisions  were  exhausted 
in  5  days:  find  the  number  of  men  in  the  reinforcement. 

27.  A  and  B  have  together  8s,  A  and  G  have  10s,  B  and  C 
have  12s.     What  have  they  each  ? 

28.  What  was  the  total  amount  of  a  person's  debts,  who  when 
he  had  paid  a  half,  and  then  a  third,  and  then  a  twelftli  of  them, 
had  still  15  guineas  to  pay? 

29.  A  father's  age  is  40  and  his  son's  8 :  in  how  many  years 
will  the  father's  age  be  triple  of  the  son's  ? 

30.  IIow  much  tea  at  4s  (Sd  must  be  mixed  with  50  lbs.  at  C^ 
that  the  mixture  may  be  sold  at  5s  CZ  ? 


CHAPTEE   IV. 

INVOLUTION   AND   EVOLUTION. 

42.  Involution  is  the  name  given  to  the  operation 
by  which  we  find  the  powers  of  quantities.  We  have 
already  (22)  had  occasion  to  notice  the  square  of  a 
binomial:  but  all  cases  of  Involution  are  merely  ex- 
amples of  Mult",  where  the  factors  are  all  the  same. 

It  should  be  noticed,  that  sitij  powe?'  of  a  power  of 
a  quantity  is  obtained  by  multij)lyiiig  together  the 
indices  of  the  two  powers. 

Thus  the  cube  of  a;',  that  is  (x^y  =  a?® ;  for  it  =  x^  ^  x^  ^  x* 
^  ^2  +  2  +  2  (20)  =  a;« :  and,  similarly,  (x^  =  a;«  =  (x^y,  that  is,  the 
square  of  the  cube  is  the  same  as  the  cube  of  the  square  of  any 
quantity,  &c. 

So  also  (a»)^=  a'^=  {a%  (2xYy='^^Y,  (-2xy'z'y=  "SxY^', 

Hence,  we  may  shorten  the  operation  of  finding 
the  4th  power  of  a  quantity  by  squaring  its  square  ; 
and,  similarly,  to  find  the  6th,  8th,  &c.  powers,  we 
may  square  the  3d,  4th,  &c. 

So  also  to  find  the  cube,  or  3rd  power,  we  may  take 
the  product  of  the  1st  and  2nd,  that  is,  of  the  quantity 
itself  and  its  square ;  to  find  the  5th,  we  may  take 
that  of  the  square  and  cube ;  to  find  the  7th,  of  the 
cube  and  4th  ;  and  so  on. 

Thus  we  shall  have 
(a  +  by  :=(a+h)  (a'  +  2ab  +  5^)  =  a'  -^  3a«6  +  3«J»  +  h\  by  Mulf, 
(a  -  by  =  (a-b)  (a^  -  2ab  +  b')  =  a'  -  3a^6  +  3a6»  -  b\ 
(a  ±  by  =  («^  ±  2ab  +  b^)  (a^±  2ab  +  b^)  =a*±  4a=&+6rt'^6^±  4a6^+  b\ 
(a  ±  by  =  (a^  ±  2ab  +  b')  (a'±  Sa^b  +  3a6^±  Z>=) 

^a'±5a*b-^lQa'h''±l0a^'^5ah^±b', 
2* 


k 


34:  INTOLUTION. 

The  above  results  should  be  remembered  and  ap- 
plied in  the  following  Examples.     The  expansions  of 
higher  powers  are  generally  best  obtained  by  the  Bi- 
nomial Theorem,  which  will  be  given  hereafter. 
Ex.  1.  (a  +  h+cy=  {a+  (b+c)y=  a'+  Za^b+c)  -^Za  (b  +  cy+  (&  +  c)' 

=a'+  3a^b  +  Za^c  +  Sab^+  Oabc  +  '6ac^+  6'+  Zb^c  +  Zbc^+  c\ 
Ex.2.  (a^b-cy^{a-(b+c)]'=a'-Za^(b  +  c)  +  Za(b  +  cy-(b-^cy 

=a'  -  Za^b  -  Za^c  +  Zab^  +  Gabc  +  Zac^  -¥-  Z¥c  -  Zbc''  -  c\ 
Or  thus : 

{a-b-cy=  {{a-b)-cY  ^  {a--by-Zia-by  c  ^  Z{a-by-c\ 
which,  of  course,  when  expanded,  would  give  the  same  result  as 
before. 

Ex.  3.  {2x  -  3)*=  {2xy-  4.3.  (2xy  +  G .  3\  (2;r)'-  4 .  Z\  (2x)  +  3* 
x=  ICj;^  -  96.r'  +  21Ca;'  -  21Ca;  +  81. 

Ex.  21. 

1.  Find  the  values  of(2^6'')»,  (-3a'5V)»,  [—^\:  (-^-^)*- 

Write  down  the  expansions  of 

2.  (.c+2)'.  Z,  {x-2y,        4.  (ic+3)».        5.  (l  +  2a:)*. 
C.  (2m-l)'.              7.  (3a; +1)*.      8.  (2a;-«)\       0.  (3a;+2a)». 

10.  (4a-3&)^  11.  (ax-yy,  12.  (ax^x^, 

13.  {2am''my,  14.  (a-5+c)'.  15.  (l-a;+ic')^ 

10.  (a  +  6aj+caj'^)».  17.  (l+aj+x^)*.  18.  (l  +  ic-ic^)*. 

10.  (l-2x+a;-)».  20.  (a-2b^cy,  21.  (l  +  2a5-3x^)*. 


43.  The  following  result  is  worthy  of  notice,  as  it 
exhibits  the  form  of  the  square  of  any  Multinomial, 
(a  +  5  +  c  +  (?  +  &€.)'=  a'+  2a  (&  +  c  +  ^  +  &c.)  +  (5  +  c  +  <?  +  &c.)' 
=  a^+  2a&  +  2rtc  +  2ad  +  &c. 

+  J^     ^2b{c  +  d    +  &c.)  +  (c  +  (?  +  &c.)* 
=  rt^  +  2a5  +  2^(5    +  2ad  +  &c. 

+  &''     +2&c+2&^  +  &c.  (i) 

+  c'  +  2c(Z    +  &c. 
+  cZ"     +  &c. 
-  «'*  +  (2rt.  +  2*)  5  +  j2  (rt  +  Z»)    +  cfc  +  J2  (a  +  &  +  c)  +  J}<Z  +  &c., 
::=«''+  (2a  +  &)  &  +  (2a'    +  c)  c  +  (2a"  +  i)^    +  &c.,  (ii) 

if  wc  write  a'  for  a  +  J.  «"  for  a  +  2>  +  c,  &c. 


INVOLUTION.  35 

We  see  from  (i)  that  the  square  of  any  multinomial 
may  bo  formed  by  setting  down  the  square  of  each 
term  and  then  the  2>'^oduct  of  the  double  of  each  term 
hy  the  sum  of  all  the  terms  thatfollovj  it. 

Another  form  of  this  result  is  given  in  (ii),  to  which 
reference  will  be  made  hereafter. 

Ex.  1.  (1  +  2a;  +  Zxy  =  1  +  2  (2a;  +  Sx^)  +  ^x^  +  4a;  (3a;'0  +  Ox' 
=  1  +  4a;  +  10a;'  +  12a;*  +  9a;*. 
Ex.2. 
{a  +  hx  +  cx'^-i-dx*  +  ex*  +  &.c.y=a^  +  2alx+2acx^  +  2adx*-t-2aex*  +  &c. 

+   h^  x^  +  2hex^i^2bdx*  +  &c. 
+     c'a;*  +  &c. 
^  aU2ah  x+  (2ac+¥)  x^-y2  (ad+hc)  xU  \2  (ae-¥l>d)  +  c^\  x*-i-&c. 
Ex.  ?>.  (1  -  2a;)«  =  (1  -  Ca;  +  12a;«  -  Sa;')' 
=  l-12a;  +  24a!'-16a;* 

+  36a;'  -  144a;'  +  9Gx' 

+  144a;*  -  192a;»  +  C4a;« 
==  1  -  12a;  +  COa;'  -  160a;'  +  240a;*  -  192a;»  +  64a;*. 
Ex.  22. 
Find  the  expansions  of 
1.  (1  +  a;  +  a;')«.  2.  (1  -a;  +  2a;'0'.  3.  (3  -2a;  +  a;")'. 

4.  (a'-2a&  +  ZP)\     5.  (2x-Zy  +  Azy,        6.  (3aa;  +  2hy  +  cz)\ 
7.  (1  -  2aa;  -  a'x'')\    8.  (2a'  -  a  -  2)\  9.  (1  -  a;  +  a;'  -  x')». 

10.  (l  +  a;)^  11.  (a;''-2a;'  +  3a;+4)'.     12.  (l-f2a;-3a;'+4a;T. 

13.  (a'  -  2a'b  +  2ah'  -  h')\  14.  (a  -  x)\ 

15.  (1  -2a;  +  3a;' - 2a;»  +  a;*)'.  16.  (a*-2a'a;+a'a;'-2aa;'+a;*)'. 

44.  Let  tlie  student  notice  the  following  remarks : 
(i)  Since  any  even  number  of  like  signs,  whether 
all  4-  or  all  -,  will  give  +  in  mult",  it  follows  that 
any  eve7i  power  of  a  quantity  is  the  same,  whether  that 
quantity  be  taken  positively  or  negatively  ;  thus,  (+a)' 
and  {-ay  are  each  =  +  a^y  and  (1  -x+xy  is  the  same 
as  \-{i-x  +  irr)]\  or  (-1  +x-xy] 

§(ii)  No  even  power  of  any  quantity  can  be  negative  ; 
(iii)  Any  odd  power  will  have  the  same  sign  as  the 


I 


36  EVOLUTION. 

45.  Evolution  is  the  name  given  to  the  operation 
by  which  we  find  the  7'oots  of  quantities. 

Since  the  G\\\)Q;power  of  a'  =  a\  therefore  the  cube 
o'oot  of  ^'  is  a?  ;  so  Vet"  —  a",  \/16a''b*  =  2a%  &c. ;  and 
so  we  may  often  extract  a  required  root  of  a  simple 
quantity,  by  dividing  it^  index  by  that  of  the  root. 

If,  however,  the  index  of  the  quantity  cannot  be 
exactly  divided  by  that  of  the  root  (as  e,  cj,  in  the  5th 
root  of  a?^  where  the  2  cannot  be  divided  by  5,)  then 
we  cannot  find  the  root  of  it ;  but  can  only  indicate 
that  the  root  is  to  he  extracted,  by  writing  down  the 
quantity,  and  the  sign  4/  before  it,  w^ith  the  index  of 
the  root  in  question ;  as  y^'',  \d.  Such  quantities 
are  called  surds^  or  irrational  quantities, 

46.  It  follows  from  (44),  that 

(i)  Any  even  root  oi  Vi  ^positive  quantity  will  have  a 
double  sign  ±  ; 

(ii)  There  can  be  wo evenrooi  of  a  negative  quantity; 

(iii)  Any  odd  root  of  a  quantity  will  have  the  same 
sign  as  the  quantity  itself. 

Thus  y V= -3j.  v-8.^v« = -2.y^  /-8r/=  ^--sf  ^^- 

Hence,  when  we  have  a  surd  expressing  an  odd  root  of  a  nega- 
tive quantity,  we  may  write  the  quantity  positive  under  the  sign  of 
evolution,  and  set  the  negative  sign  outside  ;  thus  \/  -x^  --  \/x^^ 
^6  _y  _  l^  =  a-  +  \jh'.  But  this  cannot  be  done  with  an  eten  root 
of  a  negative  quantity,  such  as  -J  -x^,  which  must  be  left  as  it  is, 
and  is  called  an  impossible  or  imaginary  quantity;  the  diifer- 
encc  between  surd  and  impossible  quantities  being  that  the  former 
have  real  values,  though  v/e  cannot  exactly  find  them,  while  there 
cannot  be  a  quantity,  positive  or  negative,  an  even  power  of  which 
would  produce  a  negative  quantity. 

Imaginary  quantities,  liowcver,  are  employed  in  some  of  the 
higher  applications  of  Algebra  ;  but  for  the  present  we  shall  leave 
the  consideration  both  of  these  and  of  surd  quantities. 


EVOLUTION.  87 

Ex.  23. 

1.  Find  the  square  roots  of  4a^h*c\  A9x*yh^,  100a"5"c*». 
9a^xY     49ajV     25a;  V" 


2.  Find  the  square  roots  of  - 


25s^    '      64a^  '     Ua'^b* 


3.  Fmdy/-^-,   4/-2^r,   |/i25^r.,    |/ 343— 


46.  To  find  the  square  root  of  a  compound  quantity. 

"We  ^7i6>i^  that  the  square  of  «  +  5  is  a'  +  2a5  +  V  ; 
let  us  see  then  how  from  a^  +  ^ah  +  5'',  we  might 
deduce  its  square  root  a-j-h. 

a^+2aJ>  +  P(a  +  h  Let  us  write  down  then  the  quantity 

a^  a^  +  2ab  +  Z>^.      Now  a,  the  first  term  of 

2a  +  h)  2ah  +  h^  the  root,   may  be  found  immediately  by 

2ab  +  &'■*  taking  the  square  root  of  its  first  term :  set 

a  then  on  the  right,  and  then  subtract  a^ ;  we  have  now  remaiu- 

ing  2a'b  +  h^j  and  if  we  divide  2ah  by  2a,  we  get  +  ^,  the  other  term 

in  the  root :  lastly,  if  we  add  this  h  to  the  2a,  multiply  the  2a  +  h, 

thus  formed,  by  h,  and  subtract  the  product,  there  is  no  remainder. 

Now  we  may  follow  this  plan  in  any  other  case,  and  if 
we  find  no  remainder,  the  root  will  be  exactly  obtained. 
Ex.  1.  Ex.  2. 

9a;-  +  6x1/  -f  y^  (3.^'  -f  y  Ua^  -  66ah  +  49&'  (4a  ~  7b 

9a;*  IGa^ 

6a;  +  1/)  Qxy  +  2/*  8a  -  7h)  -  56a5  +  49&* 

6xy  -f  y''  -  56ab  ^  493>* 

Ex.  3.  4a^  -  4ab  -  5*  (2a  -  b 

4a^ 
4a  -h)-  4a.b  -  b^ 
-  4ab  -f  l^ 
_  -2b^ 

Here  we  find  a  remainder  -  2b^ ;  we  conclude,  therefore,  that 
2a-  J  is  710 1  the  exact  root  of  4a^  ~  4ab  -  b^^  which  is  a  surd,  and 


can  only  be  written  Vl«*^-  4a?;  -  b^. 


I 


8S  EVOLUTION. 

Ex.  24. 

Find  the  square  roots  of 
1.  Ax^  +  4xy  +  y\  25a''  -  SOah  +  Ob\  25a;*  +  ZOx^y  +  9x*y\ 
2.4:9a^h'-Ua^b+a\   16a;y +40aJ2/'s+25i/V,  25a*5V  +  10»'5c'+cl 


47.  If  the  root  consist  of  more  than  two  terms,  a 
similar  process  will  enable  us  to  find  it,  as  in  the  fol- 
lowing Example  ;  where  it  will  be  seen  that  the  divi- 
sor at  any  step  is  obtained  by  doubling  the  quantity 
already  found  in  the  root,  or  (which  amounts  to  the 
same  thing  and  is  more  convenient  in  practice)  by 
doubling  the  last  term  of  the  preceding  divisor^  and 
then  annexing  the  new  term  of  the  root. 

Ex.  16a;«-24a;'  +  25aj*-.20aj'  +  IQx'-^x  +  1  (4x«-  Zx"  +  2a;-l 

^x^-Zx')-2ix''  ^25x' 


^x^-.Qx'  +  2x)  16a;* -20a;*  +  10a;* 
iea;*-12a;'+    4a;* 


8a;^-G.7;*  +  4a;-l)-8a;*+    6a;*-4a;+l 
-  8a;'  +    6a;'  -  ^'aj  +  i 


48.  The  reason  of  the  above  method  may  be  thus 
exhibited  by  considering  the  square  oi  a  +  h  +  c. 

<i*+  2ab  +  5*  +  2ac  +  25c  +  c*  (a-  +  J  +  c 

2a  +  h)    2ah  +  l^ 
2ab  +  h^ 


2a'  +  c  =  2a    +  2&  +  c )  2ac  +  2hc  +  c* 
2ac  +  21)c  +  c* 


Here  we  find,  as  before,  the  first  two  terms  in  the  root,  «  +  6, 
subtracting  first  a*,  and  then  2ab  +  i^,  that  is.  in  all,  a'  +  2ah  +  5' 
or  {a  +  hy.  Now  denote  a  ^  h  hy  a\  so  that  {a  +  h  +  cy=  (a'  +  cY 
-  a'^+  2a' c  +  c* ;  then  we  see  that,  at  this  stage  of  our  progress,  we 
have  found  a  in  the  root,  and  have  subtracted  a'^.  and  therefore 


EVOLUTION.  39 

our  remainder  will  be  no  other  than  2a'c  +  c^.  [We  see,  in  fact, 
that  2ac  +  2hc  +  c^  =  2  (a  +  h)  c  +  c^  =  2a'c+c^.']  Just  in  the  same 
way  then  as  when,  having  found  a  and  subtracted  a',  we  took  2a 
for  our  trial-divisor  in  order  to  find  5,  dividing  by  it  the  first  term 
of  the  first  remainder  2ab  +  &*,  so  now  we  take  2a'  for  our  trial- 
divisor,  in  order  to  find  c,  dividing  by  it  the  first  term  of  the 
second  remainder  2a'c  +  c'.  We  may  get  2a'  or  2a  +  25,  by  merely 
doubling  the  last  term  of  the  preceding  divisor ;  and  now  sub- 
tracting 2a' c  +  c"^,  we  shall  have  subtracted  in  all  a'*  +  2a' c  +  c\ 
that  is,  the  square  of  a  +  5  +  c. 

In  like  manner,  if  the  root  were  a  +  !>  +  c  +  d,we  may  find 
a  +  &  +  c  as  before,  and  put  it  =  a"  :  then  (a  +  1  +c  +  dy=(a"+d)* 
=  a"^  +  2a"d+d'^,  and,  as  we  shall  have  already  subtracted  (a +J^+c)' 
or  a"^,  the  third  remainder  will  be  2a"d+d^ ;  and,  therefore,  tak- 
ing 2a"  as  trial-divisor  (obtained  as  before  by  doubling  the  last 
term  of  the  preceding  divisor  2a  +  2b  +  c),  we  may  get  ^,  &c. 

It  will  be  seen  that  the  successive  subtrahends  in  the  above 
operation  are  a'^^  (2a  +  &)  5,  (2a'  +  c)  c,  (2a"  +  d)d,  &c.,  and  of 
course,  the  sum  of  them  all,  that  is,  the  whole  quantity  sub- 
tracted, is  (43  ii)  («  +  5  +  c  +  ^  +  &c.)^. 

49.  As  the  4:th  jpoive7\of  a  quantity  is  the  square  of 
its  square  (4^),  so  the  4th  root  of  a  quantity  is  the 
square  root  of  its  square  root,  and  may  therefore  be 
found  by  the  preceding  rule. 

Thus,  if  it  be  required  to  find  the  4th  root  of  a*  +  4a^x  +  (ja-x* 
+  Aax^  +  a;*,  the  square  root  will  be  found  to  be  a'  +  2ax  +  x^,  and 
the  square  root  of  this  to  be  a  +  ic,  which  is  therefore  the  4th  root 
of  the  given  quantity. 

50.  It  should  be  noticed  as  in  (45)  that  all  eve^i  roots 
have  double  signs^ 

Thus  the  square  root  of  a^  +  2a'b  +  5^  may  be  -(a  +  J),  that  is, 
-a -5,  as  well  as  ^  +  &:  and,  in  fact,  the  first  term  in  the  root, 
which  we  found  by  taking  the  square  root  of  a*,  might  have 
been  -  <x  as  well  as  a,  and  by  using  this  we  should  have  obtained 
also  -  K 

So  in  46,  Ex.  1,  the  root  may  also  be  -  3.t  -  y ;  in  47, 
-  4ic^  +3ic^  -2rc  +  1 ;  and  in  all  these  cases  we  should  get  the  two 
roots  b}'  givincc  a  double  sig:n  to  the  first  term  in  the  root. 


II 


40  EVOLUTION. 

Ex.  25. 

Find  the  square  root 
1.  Of  l+4ic  +  10a;^  +  12a;^  +  9a;\        2.  Of  9a;<  +  12aj'+22a;^  +  12a;+^. 

3.  Of  da""  +  12ah  +  W  +  6ac  +  45c  +  c\ 

4.  Of  X*  -  Sx^y  +  24x^2/'  -  32aJ2/'  +  '^^V'- 

5.  Of  4a*  -  12ft'  +  25a''  -  24a  +  IG. 

6.  Of  16a;*-lGa&a;^  +  16&V  +  4a-5'-8aZ^''  +  46*. 

7.  Of  a;«  -  4x'  +  10a;*  -  20a;*  +  2a;'*  -  24a;  +  IG. 

8.  Of  9a^  -  Gah  +  30«c  +  Qad  +  6^-  106c-  26^  +  25c*  +  lOcd  +  dr 

9.  Of  a;'  -  4a;>  +  8a;*i/^  -  lOa;'^/'  +  ^^'2/'  -  4a;y'  +  y". 

10.  Of  1  -  6a;  +  15a;^  -  20a;'  +  15a;*  -  Gx'  +  x\ 

11.  Of  4  -  12a  +  5a^  +  14a'  -  11a*  -  4a'  +  4a«. 

12.  Of  j9^  +  2pqx  +  (2j9r  +  q"")  a;^  +  2  (p«  +  qr)  a;'  +  (2q8  +  r")  x* 

+  2rsx^  +  s^a;®. 
Extract  the  4th  root 

13.  Of  1  -  4a;  +  Ga;^  -  4a;'  +  x',  and  of  a*  -  8a'  +  24a^  -  32  a  +  IG. 

14.  Of  IGa*  -  9Ga'6  +  21Ga^6^  -  21Ga6'  +  816*. 

Extract  the  8th  root 

15.  Ofa;«-lGa;'  +  112a;''-448a;'  +  1120a;*-1792a;'  +  1792a;M024a;+25G. 
IG.  Of  a^-8a^6  +  28a^6=-5Ga'6'  +  70a*6*-5Ga'6'  +  28a''6»-8a6'  +  h\ 

51.  The  inetliod  of  finding  the  square  root  of  a 
numerical  quantity  is  derived  from  the  foregoing. 

Since  l=r,  100=10^  10000  =  100^  &c.,  it  follows 
that  the  square  root  of  any  number  between  1  and  100 
lies  between  1  and  10,  that  is,  the  square  root  of  any 
number  having  one  or  tioo  figures  is  a  number  of  one 
figure;  so  also  the  square  root  of  any  number  between 
100  and  10000,  that  is,  having  thr^ee  or  four  figures, 
lies  between  10  and  100,  that  is,  is  a  number  of  two 
figures,  &c.  Hence,  if  we  set  a  dot  over  every  other 
figure  of  any  given  square  number,  beginning  with 
the  units  figure,  the  number  of  dots  will  exactly  indi- 
cate the  number  of  figures  in  its  f=^quare  root. 

a  be 

Ex.          186624  (400  +  30  +  2 
100000 a^ 

(2a  +  6) 800  +  30  =  830)   26624 

24900 (2rt  +  b)b 

(2a'+  c) 800  +  60  +  2  =  862)  1724 

1724 (2a'  +  c)f 


EVOLUTION.  4i 

Here  the  number  of  dots  is  three,  and  therefore  ihe  number 
of  figures  in  the  root  will  be  three.  Now  the  greatest  square- 
number,  contained  in  18,  the  first  period  (as  it  is  called),  is  16,  and 
the  number  evidently  lies  between  160000  and  250000,  that  is, 
between  the  squares  of  400  and  500.  "We  take  therefore  400  for 
the  first  term  in  the  root,  and  proceeding  just  as  before,  we  obtain 
the  whole  root,  400  +  30  +  2  =  432.  The  letters  annexed  will  indi- 
cate how  the  difierent  steps  of  the  above  correspond  with  those 
of  the  algebraical  process  in  (48),  from  which  it  is  derived. 

Ex.  1.  The  cyphers  are  usually  omitted  in  practice, 

186624  (432      and  it  will  be  seen  that  we  need  only,  at  any 

step,  take  down  the  next  period,  instead  of 

K^^  the  whole  remainder. 

gQ2)i724  Ii^  Ex.  2,  notice  (i)  that  the  second   re- 

1724  mainder  49  is  greater  than  the  divisor  47  ; 

Ex.  2.  this  may  sometimes  happen,  but  no  difficulty 

7784 i  (279     can  arise  from  it,  as  it  would  be  found  that  if 
4  instead  of  7  we  took  8  for  the  second  figure, 

47)378  the  subtrahend  would  be  384,  which  is  too 

329 
: large :  And  (ii),  that  the  last  figure  7  of  the 

4941  ^^^^  divisor,  being  doubled  in  order  to  make 

- — ~  the  second  divisor,  and  thus  becoming  14, 

10291^64  (3208  ^^"^^^  ^  *^  ^^  2lMq^  to  the  preceding  figure,  4, 

9"  which  now  becomes  5.    In  fact  the  first  di- 

62)129  visor  is  400 +70,  which,  when  its  second  term 

124^  is  doubled,  becomes  400  +  140  or  540. 

6408)51264  In  Ex.  3,  we  have  an  instance  of  a  cypher 

r  1  OCA. 

*iJi_  occurring  in  the  root. 

52.  If  the  root  have  any  number  of  decimal  places, 
it  is  i^lain  (by  the  rule  for  the  mult"  of  decimals)  that 
the  square  will  have  twice  as^many,  and  therefore  the 
number  of  decimal  places  in  every  square  decimal  will 
be  necessarily  even^  and  the  number  of  decimal  places 
in  the  root  will  be  half  that  number.  Hence,  if  the 
given  square  number  be  a  decimal,  and  therefore  one 
of  an  even  number  of  places,  if  we  set,  as  before,  the 
dot  upon  the  units-figure^  and  then  over  every  other 


42  irv^oLUTiox. 

figure  on  loth  sides  of  it,  the  number  of  dots  to  the 
left  will  still  indicate  the  number  of  integral  figures 
in  the  root,  and  the  number  of  dots  to  the  right  the 
number  of  decimal  places. 

Thus  10.291264  would  be  dotted  10.291264,  the  dot  being  first 
placed  on  the  units-figure  0  ;  and  the  root  will  have  one  integral 
and  three  decimal  places,  that  is,  would  be  (Ex.  3  above)  3.208. 

If,  however,  the  given  number  be  a  decimal  of  an 
odd  number  of  places,  or  if  there  be  a  rem''  in  any  case, 
then  there  is  no  exact  square  root,  but  we  may  ap- 
proximate to  it  as  far  as  we  please  by  dotting  as  before, 
{^remembering  to  set  the  dot  first  u/pon  the  units  figure^) 
and  then  annexing  cyphers  (which  by  the  nature  of 
decimals  will  not  alter  the  value  of  the  number  itself) 
and  taking  them  down  as  they  are  wanted,  until  we 
have  got  as  many  decimal  places  in  the  root  as  we 
desire. 

Ex.  Find  the  square  roots  of  2  and  259.351,  to  three  decimal 
places. 

Ex.  1.     2  (1.4U  <fcc.  Ex.  2.     259.3510  (16.104  &c. 

1  1 

24)100  ^         26)159 

96  156 

281)400  821)335 

281  321 

2824)1 1900  82204)141000 

11296  128816 


Ex.  26. 

Find  the  square  roots 

1.  Of  177241,  120409,  4816.36,  543169,  1094116,  18671041. 

2.  Of  4334724,  437.6464,  1022121,  408.8484,  16803.9369. 

3.  Of  14356521,  5742.6084,  229.704336,  74684164,  4888521. 

4.  Of  17.338896,  69355584,  6595651.24,  129208689,  975312900. 

5.  Of  16.353936,  65415744,  25553025,  43996689,  229977225. 

6.  Extract  to  five  figures  the  square  roots  of  2.5,  2000,  .3,  .03, 

111,  .00111,  .004,  .005. 


EVOLUTION.  43 

53.  Tojmd  the  cicbe  root  of  a  compound  quantity. 
Let  us  consider  the  quantity  a^  +  Za^h  +  ZaV  +  l\  ■ 

which  we  know  to  be  the  cube  of  a  +  5,  and  see  how 

we  may  get  the  root  from  it. 

a'+3a'5  +  3aJ'  +  2'^  (a +  5    Wc  may  get  a^  as  before,  by  merely 

a*  taking  the  cube  root  of  the  first  term  a' ; 

3a')  3a^5  +  3a&'*  +  &'         then,  subtracting  a^,  wchave  a  remainder 

Za''h^?>abUV         Sa-h  +  Zah""  +  P:  by  dividing   the   first 

term  of  this  remainder  by  3a^,  we  shall 

get  h,  the  other  term  in  the  root,  and  then,  if  we  subtract  the 

quantity  3a^5  +  Zah^  +  l^y  there  will  be  no  remainder. 

Pursuing  the  same  course  in  any  other  case,  if  there 
be  no  remainder,  w^e  conclude  that  we  have  obtained 
the  exact  cube  root. 

Here  the  quantity  corresponding 

Sx^+12x'^i/  +  6xy^+y^  (2x+y     to  the  trial-divisor  3a^  is  3  (2xy 

8ic'  =  12.i?',  that  to  Za'^b  is  12x''y,  that  to 

12a;^)  12x^y  +  Gxy^  +  y^  ^aV  is  Ga-y',  and  that  to  h^  is  y^ ;  so 

1 2x'^y  +  (Sxy^  +  ?y'  that  the  whole  subtrahend  is 

\2x'^y-^(jxy'^  +  y*. 
By  attending  however  to  the  following  hint,  the  subtrahend 
may  be  more  easily  constructed. 

a'+  Zo}h  4-  Zal^-^-  b^  (a-^h 
a' 
Za  +  h    3a* 

(3a+J)J 


Za^  +  aah  +  P 


Za^h  +  Zah'-^  h' 
Za^h  +  Zah''+  h* 


Set  down  first  3a,  some  little  way  to  the  left  of  the  first  re- 
mainder, and  then,  multiplying  this  by  a,  obtain  3a'  as  before  ;  by 
means  of  this  trial-divisor  find  i,  and  annex  it  to  the  3a,  so  making 
Za+b]  multiply  this  by  J,  and  set  the  product (3a +  2»)  b  or  Zab  +  b* 
under  the  3a'j  and  add  them  up,  miking3a'  +  3a5  +  6';  then,  mul- 
tiplying this  by  5,  we  have  Za^b  +3aZ»'-f  P,  the  quantity  required. 

Tho  value  of  the  above  method,  in  saving  labour,  will  be  more 
fully  seen  when  the  root  has  more  than  two  terms,  or,  if  numeri- 
?\  more  than  two  figures. 


^^S  iJ^iore  thai 


u 


Ex. 

6x  +  y    12x' 

+ 

Gxy 

KyOLU-nON. 

8a;''  +  I2x''y  +  Qxy^  +  y^  (2x  +  y 

12x^y  +  6.T^^  +  2/' 

12aj^ 

+ 

Oxy 

+  y' 

12ic^y  +  6xy^  +  y* 

Ex.  27. 

Find  the  cube  roots 
1.  Of  x^+6x''y  +  12xy''  +  SyK  2.  Of  a^-9aU27a-27. 

3.  Of  a;*+12jj^+48ic  +  G4.  4.  Of  Sa' -ZOa'h^ 6 iah'' -27 h\ 

5.  Of  a'+24a^Z>+192a6^  +  5126^     G.  Of  8^'-84a;V+294iC2/'-343i/^ 

7.  Of  m^  -  12m''nx  +  48/?i;.V  -  G4/iV. 

8.  Of  oV  -  I5a''hx'  +  75ab''x^  -  I25b'x\ 


54.  If  the  root  consist  of  more  than  two  terms,  as  a  +  i>  +  c,  we 
may  (just  as  in  the  case  of  the  square  root)  first  find  a+6  as  above, 
and  put  this  =  a' :  then,  at  this  point  of  the  operation,  we  shall 
have  subtracted  first  a^  and  then  3a'^b  +  Zab^  +  b\  that  is,  altogether  . 
(a  +  by  or  a'^ ;  and  therefore,  since  the  whole  quantity  (a  +  &  +  c)' 
=(a'+c)'=a''+3a'^c  +  3a'c^  +  c',  the  remainder  will  be  no  other  than 
Sa'^c  +  3a'c'  +  c^  [Tn  fact,  as  was  done  in  the  case  of  the  square 
root,  it  may  be  easily  shewn  to  be  identical  with  this.]  If,  there- 
fore, we  take  now  as  trial-divisor  3a'*,  just  as  before  we  took  3rt'*, 
we  shall  get  c  the  third  term  in  the  root,  and  subtracting  the 
quantity  Sa'^c  +  ^a'c^  +  c^,  we  shall  have  no  more  remainder. 

Now  the  process  of  finding  3a  ^  is  much  simplified  by  observing 
that  it  =3  {a+by=Za^  +  Oab  +  W  \  but,  if  we  add  b\  the  square  of 

the  last  term  in  the  root,  to  the  two  lines  „  ,    o  i      L  ?  the  sum 
'  da^  +  oab  +  0^ 

will  be  exactly  3a^  +  GaZ)  +  3Z/',  the  quantity  required.  By  this 
means  then  we  get  3a"-',  and  then  have  only  to  set  to  the  left  of  it 
3a'  or  3a  +  36,  (which  may  be  found  by  tripling  the  last  term  of 
the  preceding  divisor  3a  +  b)  and  proceed  just  as  we  did  befoi*e 
when  we  had  set  down  3a  and  3a* — that  is,  first  finding  c,  and 
then  forming,  as  before,  3a'-c  +  3a'c*  +  c',  which  we  subtract, 
making,  with  a''  already  subtracted,  (a  +  cf  or  (a  +  h  +  c)^  sub- 
tracted altogether.  And  so  on,  if  the  root  were  a  +  b  +  c  +  d^  &c 
The  student  should  study  carefully  the  first  of  the  two  follow- 
ing Examples,  as  it  is  the  type  to  which  all  others  .are  reft-rred. 


EVOLUTION. 


45 


I 

CO 


+  . .    W 

CO  ^    t^ 

^*  ^     K^ 

•*  CO  <D 

5?-  «.    « 


CO    f-f 


00   g. 

o*  CO  tr 
»    5^    o 

W.         O         Hi 

CO  uQ 
+      §' 


CQ 


I- 


CO 


CO 
CO    O      -^ 


'    O 

B 


CO  g. 

-r      &  S' 

^       I 

CO  M  ^ 

'    +  ^ 

55  *^ 


I 

CO  -  >_, 

I  O     "-J 

to  CO    ^ 

sr  »   >-• 


r^ 


v.^  CO    ^ 

CO  ^  5* 

H.  I 

I  CO  ttj 

CO  c* 

»>  o  hQ 

CO  ^,  H. 

p  -•  - 


ii 


1 

? 

1 

H 

1 
to 

1 

1 

CO 

1 

CO 

1 

t 

c 

t 

1 

C 

1 

1 

+ 

+ 

+ 

c 

c 

C 

s^ 

»» 

+ 

+ 

? 

? 

j 

+ 

+ 

^ 

^ 

1 

CO 
+ 

1 

CO 

"j~ 

1 

1 

f 

CO 

CO 

+ 

+ 

1 

+ 

>— 1 

h- ' 

«u 

t— ' 

+ 

+ 

1 

1 

1 

1 

1 

CO 

1 

00 

I 

CO 

I 

CO 


f 

00 

I 
I 

to 


CO 


CO 
Si 


CO 


CO 
Si 


CO 


CO  Ci 

II 


+ 

CO 


CO 

Si 


+ 

CO 


CO 


CO 


+ 

CO 


CO 

Si 


CO 


+ 


CO 
Si 


+ 

CO 


+ 


CO 


+ 

CO 


40  EVOLUTION. 

Ex.  28. 
Extract  the  cube  roots 

1.  Of  a''  +  6a^  +  15a*  +  20a'  +  15a^  +  Ga  +  1. 

2.  Of  x'  -  12a;^  +  6ix*  -  112x'  +  108x'  -  ASx  +  8. 

3.  Of  a'  -  Sa'b  +  Oa*b''  -  7a'b^  +  Ca^5*  -  Zah"  +  5«. 

4.  Of  aj°  -  12ax'^  +  60aV  -  IGOa^aj'  +  240aV  -  I02a'x  +  64a«. 

5.  Of  8x^  +  ASx'y  +  60xY  -  ^^^Y  -  ^OxY  +  lOSxif  -  27y\ 

6.  Of  x'  -  Zx^  +  ^x'  -  10a;"'  +  12a;*  -  12a;*  +  103;=*  -  ^x"  +  3a;  -  1. 

7.  Of  a^  -  &'  +  c'  -  3  (a-5  -  a\  -  ay  -  ac"  -  Vc  +  W)  -  ^abc. 

8.  Of  l-6a;+21a;^-56.t;=  +  llla;*-174a;*+219a;«-204a;U144a;«-64a;». 


55.  Since  1  ==  1^  1000  =  10^  1000000  =  100',  &c., 
it  follows  that  the  cube  root  of  any  number  between 
1  and  1000,  that  is,  having  one^  two^  or  three  figures, 
is  a  number  of  07ie  figure ;  so  also  tlie  cube  root  of 
any  number  between  1000  and  1000000,  that  is,  having 
fouT^five^  or  six  figures,  is  a  number  of  two  figures, 
&c.  Hence,  if  we  set  a  dot  over  every  tliird  figure  of 
any  given  cube  number,  beginning  with  the  units- 
figure,  the  number  of  dots  will  exactly  indicate  the 
number  of  figures  in  its  cube  root. 

If  the  root  have  any  number  of  decimal  places,  the 
cube  will  have  thrice  as  many ;  and  therefore  the  num- 
ber of  decimal  places  in  every  cube  decimal  w^ill  be 
necessarily  a  multiple  of  three^  and  the  number  of 
decimal  places  in  the  root  will  be  a  tliird  of  that  num- 
ber. Hence,  if  the  given  cube  number  be  a  decimal, 
and  consequently  have  its  number  of  decimal  places  a 
multiple  of  three,  by  setting  as  before  the  dot  upon 
the  units-figure^  and  then  over  every  third  figure  on 
l)oth  sides  of  it,  the  number  of  dots  to  the  left  will  still 
indicate  the  number  oi integral  figures  in  the  root,  and 
the  number  to  the  right  the  number  of  decimal  places. 

If  the  given  number  be  not  a  perfect  cube,  we  may 
dot  as  before,  (always  setting  the  dot  first  upon  the  units- 


EVOLUTION. 


47 


figicre)y  and  annex  cyphers  as  in  the  case  of  the  square 
root,  so  as  to  ai)proximate  to  the  cube  root  required, 
to  as  many  decimal  places  as  we  please. 

It  will  be  seen,  by  the  following  example,  how  the  numerical 
process  corresponds  to  the  algebraical.  The  cyphers  are  omitted, 
except  that  in  the  numbers  corresponding  to  3tt*,  3«j'',  &c.,  it  is 
necessary  to  express  two  at  the  end :  thus  a  is  really  4000,  and 
therefore  Za^  is  48000000  ;  but  as  in  the  first  remainder  we  only 
need  the  figures  of  the  first  and  second  periods,  corresponding  to 
43  in  the  root,  we  may  treat  the  a  as  40,  and  thus  3a'  will  be 
4800  and  Za  will  be  120,  so  that  Za  +  l  will  become  123. 


Ex. 


80677568161 (4321 
64 

4800 
369 

5169 

16677 
15507 

554700 
2584 

1170568 

557284 

1114568 

55987200 
12961 

56000161 

56000161 

56000161 

123 


1292 


12961 


Ex.  29. 
Find  the  cube  roots  of 

1.  9261,  12167,  15625,  32768, 103.823,  110592,  262144,  884.736. 

2.  1481544, 1601.613,  1953125,  1259712,  2.803221,  7077888. 

3.  12.812904,  8741816,  56.623104,  33076.161,  22425768. 

4.  102503.232,  820025856,  264.609288,  1076890625,  2.116874304. 

5.  Extract  to  4  figures  the  cube  roots  of  2.5,  .2,  .01,  4. 


CIIAPTEE   V. 

GREATEST  COMMON  MEASURE  I    LEAST  COMMON  MULTIPLE. 

56.  When  one  quantity  divides  another  without 
remainder,  it  is  said  to  measure  it,  and  is  called  a 
rtieasure  of  it. 

ThuSj  3,  «,  J,  3g^j  ab,  a^^  &c.  are  all  measures  of  Za^h. 

A  common  measure  of  two  quantities  is  one  which 
divides  each  of  them  without  remainder. 

Thus,  a,  &j  3^5  35,  ab^  3a5,  are  all  common  measures  of  Za/h  and 
V^abc ;  and  their  greatest  common  measure,  that  is,  the  largest 
common  factor  they  contain,  is  3a5. 

57.  It  is  commonly  easy  to  detect  Tjy  inspection^  i,  e. 
by  looking  at  the  two  quantities,  their  largest  common 
measure,  if  it  is  a  simple  factor,  that  is,  if  it  consists 
of  only  one  term  ;  because  then  it  will  be  found  as  a 
factor  in  every  term  of  each  of  them. 

Thus,  Zxy  will  divide  every  term  of  Zx^y  -  Gxy^  and  also  of 
Zxy-9x^y' ;  it  is  therefore  a  common  measure  of  them :  and  since 
when  these  are  divided  by  Zxy^  the  quotients  cc^  -  2i/'  and  1  -  Zxy 
have  no  common  factor,  Zxy  is  their  greatest  common  measure 
(g.  c.  m.). 

So  2a^Z>  is  the  greatest  divisor  of  (ja'h^  -  8a%  and  a-c  of 
2a'c'  —  5a^hc ;  and  a^,  which  is  the  g.  c.  m.  of  2a^b  and  a^c,  is 
plainly  therefore  the  g.  c.  m.  of  Oa^b^  -  Sa*b  and  2a'c'  -  5a'6c. 
Ex.  30. 

Find  the  g.  c.  m.  of 

1.  3a;'  and  I2x^y ;  4:a^b^  and  -  6ab^ ;  -  I2x'yh*  and  8i/zK 

2.  Zax^  -  ^a'x  and  a V  -  Zabx ;  3a»  +  la'b  -  5a5'  and  2a'6  +  2aV  ; 

Caj'y  ~  12x^y^  +  Zxy^  and  4aa;'  +  4:axy  +  4a'a;. 


58.  In  like  manner  we  may  sometimes  find  by 
inspection  the  g.  c.  m.  of  two  quantities,  when  not  a 
simple  factor,  if  it  happens  to  be  easy  to  separate 
them  into  their  component  factors. 


GREATEST   COMMON   MEASURE.  49 

Ex.  1.  The  G.  c.  M.  of  GaV  (a*  -  x^)  and  Aa^x  (a  +  xf  is 
20^ X  (a  +  x). 

Ex.  2.  The  g.  c.  m.  of  a"  (ci'x''-  Zax^  +  2x')  and  a;'  (a*  -  4«V), 
that  is,  of  a^x"^  (a*  -  Zax  +  2x'^)  or  a^x^  (a  -  2x)  (a  -  x)  and 
a'x^  (a''  -  4x'),  is  ^^o;^  (^  -  2x). 

Ex.  31. 

Find  by  inspection  the  g.  c.  m.  of 
1.  4x^  (a+xy  and  10  (a^'x-xy.        2.  o)^  (a^-x^y  and  (a'*a;+aic^)». 
3.  (a'b-aPy  and  «&  (a^-5^)^  4.  6  (x^-1)  and  8  (iC^-3iC+2). 

5.  (;i'^ + 0.-)'  and  ic^  (x''-x-2).  C.  4  (o;^  +  a^)  and  6  (a;'-2aa;-3a»). 

7.  a'  (a;^  -f  12^  +  11)  and  aV  -  Ua"x  -  12a\ 

8.  9  (aV  -  4)  and  12  (a'^x''  +  4aaj  +  4). 


59.  But  if  the  greatest  common  measure  of  two 
quantities  be  a  compound  quantity^  it  cannot  generally 
be  thus  easily  found  by  inspection,  but  may  always 
be  obtained  by  a  method  we  are  now  about  to  ex- 
plain, the  proof  of  which  will  be  given  hereafter. 

Def.  An  algebraical  quantity  is  said  to  be  of  so 
many  diw^nsions^  as  is  indicated  by  the  highest  index 
of  its  letter  of  reference. 

Thus  aj^  -  7ic  +  10  is  of  tico  dimensions,  aj^  +  1  of  three. 

If  it  also  involve  other  letters,  it  is  said  to  be  of  so 
many  dimensions  in  each  of  them,  according  to  the 
highest  indices  of  each. 

Thus  x*y+Sx^y^+x^y^  is  of  four  dimensions  in  x,  and  three  in  y. 

If  the  dimensions  of  each  term  are  the  same,  the 
quantity  is  said  to  be  homogeneous,  and  of  so  many 
dimensions  as  is  indicated  by  the  sum  of  the  indices 
in  each  terra. 

Thus  the  last  quantity  is  homogeneous^  and  of  five  dimensions. 

The  word  dimensions  has  been  adopted  from  the  language  of 
Geometry ; — such  quantities  as  «,  6,  &c  being  compared  to  lines 
(which  have  only  one  dimension,  viz.  length),  and  called  linear 
3 


60  GREATEST  COM^ION  MEASURE. 

quantities ;  such  quantities  as  a?^  ab^  &c.  to  areas  (which  have  two 
dimensions,  length  and  breadth)  ;  and  such  as  a^  a"^!)^  ahc,  &c.  to 
solids  (which  have  three  dimensions,  lengtli,  breadth,  and  thick- 
ness) :  beyond  this  we  have  no  corresponding  quantities  in  Geom- 
etry ;  but  the  term  dimensions,  having  been  once  employed  in 
Algebra,  has  been  retained  in  all  other  cases. 

60.  Let  there  be  given  then  two  algebraical  quan- 
tities, of  which  it  is  required  to  find  the  g.  c.  m.  Ar- 
range them  according  to  powers  of  some  common 
letter,  and  divide  the  one  of  higher  dimensions  by  the 
other ;  or  if  the  highest  index  happen  to  be  the  same 
in  each,  take  either  of  them  for  dividend.  Take  now, 
as  in  Arithmetic,  the  remainder  after  this  division  for 
divisor,  and  the  preceding  divisor  for  dividend,  and 
so  on  until  there  is  no  remainder :  then  the  last  divisor 
will  be  the  g.  c.  m.  of  the  two  given  quantities. 

Ex.  Find  the  g.  c.  m.  ofx"  -  7ic+10  and  Ax""  -  25x^  +  20a;  +  25. 
x'-7x+10)4cx'-25x'' +  20x  +  25(Ax  +  3 
4x^  -  28a;^  +  40x 


^x^  -  20aj  +  25 
3iu'  -  21a;  +  30 


5)a;'»-7a;  +  10Cr-2 
x^-5x 


-2a; +  10 
Ans,  a;  -  5.  -  2a;  +  10 

We  may  as  well  observe,  that  the  expression  Greatest  c.  m., 
which  has  beeji  adopted  from  Arithmetic,  must  be  understood  in 
Algebra  as  applying  not  to  the  numerical  magnitude,  positive  or 
negative,  of  the  quantity,  but  to  its  dimensions  only,  on  which 
account  it  is  sometimes  called  the  Highest  c.  m.  Thus  it  would 
be  quite  immaterial  whether,  in  the  above  example,  we  consider 
the  G.  c.  M.  to  be  a;  -  5  or  5  -  a; :  and  either  of  these,  in  fact,  might 
be  made  numerically  greater  than  the  other,  by  giving  different 
values  to  x. 


GREATEST   COMMON   MEASURE.  61 

Ex.  32. 
Find  the  g.  c.  k, 

1.  Of  3.^'  +    ic  -2  and    3a)'  +    4ic  -  4. 

2.  Of  6a;'  +7x  -S  and  12a;'  +  16a; -  3. 

3.  Of  9a;'  -  25  and  Ox''  +  Sx  ~  20. 

4.  Of  8a;'  +  14a;  -  15  and  8a;'  +  30a;'  +  13a;  -  30. 

5.  Of  4a;'  +    3.^  -  10  and  4a;'  +  7a;'  -  3a;  -  15. 

6.  Of  2a;*  +  a;'  -  20a;'  -  7a;  +  24  and  2a;*  +  3a;'  -  13a;'  -  7a;  +  15. 


61.  If  the  given  quantities  have  both  or  either  of 
them,  in  any  case,  simple  factors,  as  in  (57),  these 
ninst  be  struck  out,  and  the  Kule  applied  to  the  re- 
sulting quantities.  Then  the  g.  c.  m.  of  these,  being 
found  as  above,  will  be  the  same  as  that  of  the  given 
ones  ;  ^except  it  should  happen  that  we  have  to  strike 
factors  out  of  hoth  of  them,  and  that  these  factors  them- 
selves  have  a  common  factor.  In  this  case  the  g.  c.  m. 
found,  as  above,  of  the  resulting  quantities,  must  be 
multiplied  by  this  common  factor,  in  order  to  produce 
that  of  the  given  ones. 

So  also,  whenever  we  convert  a  remainder,  accord- 
ing to  the  Eule,  into  a  divisor,  we  may  strike  out  of  it 
any  simple  factor  it  may  contain.  Here,  however, 
there  is  no  restriction,  as  in  the  former  case  ;  because 
no  part  of  such  a  simple  factor  can  be  common  also  to 
the  new  dividend,  which,  being  the  same  as  the 
former  divisor,  will  be  already  clear  of  simple  factors. 
It  is  only  with  \\\q  first  pair,  or  gi'ven  quantities^  that 
we  shall  have  to  attend  to  this. 

And  if,  moreover,  the  first  term  of  any  such  re- 
mainder is  negative,  we  may,  for  the  sake  of  neatness, 
before  taking  it  as  a  new  divisor,  change  the  signs  of 
all  its  terms,  which  is  equivalent  to  dividing  it  by  - 1* 
Tliis  can  only  affect  the  signs  of  the  g.  c.  m. 


52  GREATEST   COMMON   MEASURE. 

Ex.  Find  the  g.  c.  m.  of 

2a;^  -  8:c^  +  12.i;'  -  Sx^  +  2x  and  Zx^  -  6:c«  +  3^. 
Here,  striking  out  of  the  first  the  factor  2x  (which  is  common  to 
all  its  terms)  and  of  the  second  the  factor  Sx,  we  reduce  the  quan- 
tities to  X*  -  4x^  +  Qx"^  -4x  +  1  and  x*-2x^  +  1;  but  as  2x  and  Zx 
have  themselves  a  common  factor,  x,  it  is  plain  that  the  original 
quantities  have  a  common  factor  x,  which  these  latter  quantities 
have  not ;  hence  the  g.  c.  m  of  these,  when  found,  must  be  multi- 
plied by  X  to  produce  that  of  the  given  quantities. 
x'-2x^  +  l)  x'-4.x^+6x''-ix+l  (1  x'-2x+l)  ic*-2aj'  +  l  (aj'  +  2a;+l 
x*-2.t^  +  l  x^-2x^+x^ 

-  4:x\-4x^+Sx^-4x  2x^-Sx^  + 1 

x''-2x  +1  2x^-4x^-^2x 

a;^-2j;+l 
ic'-2j+l 

In  this  Example,  the  first  remainder  is  reduced  by  dividing  it  by 
^4x ;  and,  the  g.  c.  m.  of  these  two  quantities  being  x"^  -  2x  +  1.  that 
of  the  two  given  quantities  will  be  x  (x"^  -  2a;  +  1)  or  x^  -2x^  +  x, 

Ex.  33. 
Find  the  c.  c.  m. 
1.  Of  a^+.^•'anda'+2«aJ  +  ic^      2.  Of  ic'+ a;-2  and  a;'- 3a;  +  2. 

3.  Of  2x^  +ex^  +  Qx  +  2  and  Ox^  +  6x^  -  ex  -  6, 

4.  Of  2if  -  lOy""  +  12y  and  3^/*  -  IS^/'  +  2hf  -  24. 

5.  Of  x^  -  eax""  +  12^=0?  -  ^a^  and  a;*  -  4aV. 

6.  Of  2x^  +  10a;'  +  14a;  +  6  and  a;'  +  a;'  +  7aj  +  39. 

7.  Of  3a;'  +  3a;''  -  15a;  +  9  and  3a;*  +  3a;'  -  21a;'  -  9a;. 

8.  Of  a;'  +  xhj  +  xy"^  +  y^  and  a;*  +  x^y  +  xy^  -  y*, 

9.  Of  2a'  +  a'b  -  A.arh''  -  Zah'  and  4a'  +  a'b  -  2a^^  +  al\ 

10.  Of  Za'+15a'b-Za'I>''-l5a'h''  and  lOa'-ZOa'b-lOa'h^-i-ZOaP. 

11.  Of  X*  -  2x^y  +  2xy^  -  y'  and  x'  -  2x^y  +  2a;'7/'  -  2xy^  +  y\ 

12.  Of  x'  +  Ca;'  +  11a;'  +  4a;  -  4  and  x'  +  2a;'  -  5a;'  -  12a;  -  4. 


62.  If  now,  having  first  attended  to  the  directions 
of  (61),  we  findj  at  any  step  of  our  process,  that  the 
first  term  of  the  dividend  is  not  exactly  divisible  by  the 
first  of  the  divisor,  then,  in  order  to  avoid  fractions  in 
the  quotient,  we  may  multiply  the  whole  dividend  by 


GREATEST   COMMON   MEASURE.  53 

Gucli  a  simple  factor,  as  will  make  its  first  term  so 

divisible. 

Ex.  Find  the  g.  c.  m.  oi(jx'^y  +  ^xy'^-2y^  and  ^x'^  +  Ax'^y-^iXy^, 
Stripping  them  of  their  simple  factorSj  2y  and  4a',  (and  noting 

that  these  contain  the  common  factor,  2),  we  have  Zx^  +  2xy  —  y^ 

and  2x^  +  xy  -  y^,  and  proceed  with  these  quantities  as  follows : 


" 

+  xy' 

Sx' 
2 

+  2xy' 

-y' 

'(3 

2x' 

-y')6x' 
ex'' 

+  4:xy-2y' 

+  Zxy-Zy' 

y\xy  +  2/' 

x  +  y) 

2x''  + 
2x''  + 

xy- 
2xy 

-y'i2x- 

-y 

- 

xy- 

-y" 

-^ 

xy- 

■y' 

The  G.  c.  M.  then  will  be  2{x  +  y\   it  being  plain  that  the 
G.  c.  M.  of  2  (Zx^  +  2xy-y^)  and  2x'^  +  xy-y'^  will  be  the  same  as 
that  of  Zx'  +  2xy  -  y'^  and  2x'^  +  xy  -  2/^  because  the  2  introduced 
into  the  first  is  no  factor  of  the  second  quantity. 
Ex.  34. 
Find  the  g.  c.  m. 

1.  Of  Gx'^  +  13^  +  6  and  8ic^  +  Caj-9. 

2.  Ofl5x^-.aj-Gand9iC^-3aj-2. 

3.  Of  6aj= -aj- 2  and  21aj3- 26  aj^  +  8a7. 

4.  Of  6^3  -  (jx''  +  2^-2  and  12.r^  -  15aj  +  3. 

5.  Of  3aj^  -  22aj-  15  and  5a;*  +  a;«-54a;2  +  18aj. 
G.  Of  3aj^  -  Zx'^y  +  ir^/^  -  2/^  and  ix^—x'^y  -  Zxy'^, 

7.  Of  ic^-So;  +  3  and  x'  +  Sx^  +  x  +  3. 

8.  Of  5x'  +  2x''  -  15a;  -  G  and  -  7x^  +  4a;'  +  21a;  - 12. 

9.  Of  20.T*  +  x^-l  and  25a;*  +  5a;^-a;-l. 

10.  Of  (jx^-xhj-Zxhf  +  Zxy^-%/  and  ^x^-?>xhj-2x''yUZxy^-y\ 

11.  Ofl2a;M2a;'y-  +  12a;V-3a;7/*&  12a;'  +  8a;V-18.i'y-6a;"^»+4a;2/*. 

12.  Of  x'  -  2^  h  x""  -  8a;  +  8  and  4a;3  -  12a;  '^  +  9a;- 1. 


63.  In  order  to  prove  the  Eule  above  given,  it  will 
be  necessary  to  shew  first  the  truth  of  the  following 
statement. 


54  GREATEST  COMlklON  MEASURE. 

If  a  quantity  c  he  a  common  measure  of  a  and  b, 
it  will  also  oneasure  the  sum  or  difference  of  any  mul- 
tiples of  a  and  b,  as  ma  ±  nb. 

For  let  c  be  contained  jp  times  in  a^  and  q  times  in 
h ;  then  a  =  ^(?,  5  =  qc^  and  m(2  ±  ^^^  =  '?^^<^  ±  n^<?  — 
{mjp  ±  ^^2')  <? ;  hence  c  is  contained  quj)  ±  n^'  times  in 
Qna  ±  7iZ>,  and  therefore  c  measures  ma  ±  7i5. 

Thus,  since  G  will  divide  12  and  18  without  remainder,  it  will 
also  divide  any  number  such  as  7  x  12  +  5  x  18,  11  x  12-3  x  18, 
12  (or  1  X  12)  +  7  X  18,  5x12-18,  &c.,^.  e.  any  number  found  by 
adding  or  subtracting  any  multiples  of  12  and  18. 

64.  To  prove  the  liule  for  finding  the  greatest  Com- 
mon Measure  of  tioo  quantities. 

First,  let  the  two  given  quantities,  denoted  by  a 
and  J,  have  neither  of  them  any  simple  factor. 

Let  a  be  that  which  is  not  of  lower  dimensions  than 
the  other;  and  suppose  a  divided  by  5,  with  qnotient 
j9  and  remainder  <?,  h  by  c^  with  quotient  q  and  re- 
mainder d^  &c. 

h)  a  {p  546)  672  (1 

ph  546 

c)h  {q  126)546(4 

qc  504 

d)  c  {r  42)126(3 

rd  126 

Then,  by  (63),  all  the  common  measures  of  a  and  h 
are  also  measures  of  a  -jpb  or  c,  and  are  therefore 
common  measures  of  5  and  c ;  and,  conversely,  all  the 
common  measures  of  i  and  c  are  also  measures  of 
pb  -^r  c  or  ^,  and  are  therefore  common  measures  of 
a  and  h :  hence  it  is  plain  that  h  and  c  have  precisely 
the  same  common  measures  as  a  and  h. 


GREATEST   COMMON   MEASURE.  55 

In  like  manner,  it  may  be  sliewn  that  c  and  d  have 
the  same  common  measures  as  1)  and  ^,  and  therefore 
the  same  as  a  and  J. 

And  so  we  might  proceed  if  there  were  more  re- 
mainders, the  quantities  a,  5,  c^  d^  &c.  getting  lower 
and  lower,  yet  still  being  such  that  a  and  Z>,  h  and  <?, 
c  and  c?,  &c.  liave  the  same  common  measures. 

But,  if  6?  divides  c  without  remainder,  tlien  d  is  itself 
the  greatest  quantity  that  divides  both  c  and  d^  that  is, 
<^is  the  greatest  of  the  common  measures  ofc  and  d^  and 
therefore  is  the  Greatest  Common  Measure  oi  a  and  h. 

ThuSj  in  the  numerical  example,  the  common  divisors  of  546 
and  G72  arc  precisely  the  same  as  those  of  12G  and  54G,  and  these 
again  are  the  same  as  those  of  42  and  126 :  but  42  is  the  g.  c.  m. 
of  42  and  126,  and  is  therefore  the  g.  c.  m.  of  126  and  546,  and 
also  of  546  and  672. 

65.  Next,  let  cc  and  5  have  simple  factors,  and  let 
a  =  aa\  h  =  I3h\  where  a  denotes  the  product  of  all  the 
simple  factors  in  cc^  and  /3  of  those  in  &,  and  a\  V  are 
the  resulting  quantities,  when  these  simple  factors  are 
struck  out:  then  a'  h\  having  neither  of  them  any 
simple  factor,  will  have  no  factor  in  common  with  a 
or^.  Now  a  or  aa'  is  made  up  only  of  the  fiictors  in  a 
and  a\  and  h  ov  13V  only  of  those  in  /?  and  h\  Hence,  if 
a  he  prime  to  /3,  (that  is,  if  a  have  no  factor  in  covimon 
with  /3)^  the  only  factors  which  a  can  have  in  common 
Avith  h  must  be  those  Avhich  a^  may  have  in  common 
with  h\  that  is,  the  a.  c.  m.  of  a  and  h  will  be  the  same 
as  that  of  a^  and  h\  But,  if  a  and  /3  have  any  com- 
mon factor,  then  this  will  also  be  common  to  a  and  J, 
besides  what  may  be  common  to  a^  and  h\  that  is,  the 
G.  c.  M.of  a  and  i  will  be  obtained  by  multiplying  the 
G.  c.  M.  of  a^  and  h'  by  the  common  factor  of  a  and  /3. 


56  GREATEST   COMMON   MEASURE. 

Hence  tliis  case  also  is  reduced  to  finding  the  g.  c.  m. 
of  two  quantities  a!  and  1)\  which  have  no  svmjple factors. 
And,  of  course,  the  above  reasoning  liolds  if  either 
a  or  yS  be  unity,  that  is,  if  one  only  of  the  given  quan- 
tities have  a  simple  factor  to  be  struck  out. 

^^,  Having  shewn  that  wo  may  strike  any  simple 
factors  out  of  the  original  quantities,  we  shall  now 
shew  that  we  may  strike  them  also  out  of  any  of  the 
remainders. 

Let  then  a' ^  V ^  represent  quantities  having  no  simple 
factors,  (either  the  original  quantities,  a^  &?,  if  they 
have  no  simple  factor,  or  else  a,  5,  reduced,  as  above) ; 
and  let  us  apply  the  Eule  to  a' ^  h\  dividing  a'  by  &', 
and  obtain  the  first  remainder  c  :  then  we  know  that 
the  G.  c.  M.  of  a'  and  V  is  the  same  as  that  of  V  and  c. 
Suppose  now  that  c  =  ^g\  wliere  7  is  a  simple  factor, 
and  c'  a  compound  quantity,  having  no  simple  factor. 
Then  c.h  made  up  of  the  factors  in  y  and  c' ;  and  V 
(having  no  5- mj^Z^  factor)  can  have  no  factor  in  common 
with  7,  and  therefore  can  have  none  in  common  with 
G  but  such  as  it  may  have  in  common  with  c' ;  that  is, 
tlie  G.  c.  M.  of  V  and  c  is  the  same  as  that  of  V  and  g\ 
And,  of  course,  tlie  same  reasoning  holds  with  the 
other  remainders. 

67.  Lastly,  if,  at  any  step  (supposing  simple  factors 
struck  out),  the  first  term  of  tlie  dividend  should  not 
be  exactly  divisible  by  the  first  of  the  divisor,  as,  for 
instance,  in  the  case  of  a'  and  h\  we  may  multiply  the 
dividend  a'  by  any  simple  factor  a',  whicli  will  make 
it  so  divisible  :  for,  since  the  divisor  V  has  no  shnj^l^ 
factor,  it  can  have  no  factor  in  common  with  a\  nor 
therefore  any  in  common  with  the  dividend  a'a\  but 
what  it  may  luxve  in  common  with  a\  that  is,  the  g.c.m. 
of  ^'  and  V  will  be  the  same  as  that  of  aV  and  V, 


LEAST   COMMON   MULTIPLE.  67 

68.  "When  one  quantity  contains  another,  as  a  divisor 
without  remainder,  it  is  said  to  be  a  multijyle  of  it ; 
and  a  common  multiple  of  two  or  more  quantities  is 
one  that  contains  each  of  them  without  remainder. 

Thus,  Qx^y  is  a  common  multiple  of  2a;',  Zxy^  6a;',  &c.,  and  any 
quantity  is  a  multiple  of  any  of  its  measures. 

Of  course,  the  least  common  multiple  (l.  c.  m.)  of 
two  or  more  quantities  is  the  least  quantity  that  can 
be  formed,  so  as  thus  to  contain  each  of  them. 

69.  To  find  the  Least  Common  MidtijpU  of  two 
quantities. 

Let  a  and  h  represent  the  two  quantities,  d  their 
G.  c.  M. ;  and  let  a-=^fd^  h^qd^  so  that^  and  q  will  have 
no  common  factor.  Then  the  least  quantity  which 
contains  p  and  q  wuU  be  j^^',  and  therefore  the  least 
quantity  which  contain s^^^:?  and  (^(^  will  \i^  jpqd^  which 
is  consequently  the  l.  c.  m.  required  of  a  and  5. 

Since  vqd^—     /     =        _.    ,  it  appears  that  the 
d  d 

L.  c.  M.  of  a  and  1)  may  be  found  by  dividing  their 

product  by  their  g.  c.  m.  ;  or,  which  is  more  simple  in 

practice,  by  dividing  eitlier  of  them  by  their  g.  c.  m., 

and  multiplying  the  quotient  by  the  other. 

The  L.  c.  M.,  however,  of  two  or  more  quantities  is  generally 
formed  by  inspection,  and,  with  a  little  practice,  there  is  no  diffi- 
culty in  this,  as  we  have  only  to  set  down,  the  factors  which 
compose  them,  omitting  any  that  may  occur  more  than  once,  and 
the  product  of  these  will  be  the  l.  c.  m.  required. 

Ex.  1.  Find  the  l.  c.  m.  of  26a;,  (Scibxy,  Zacx, 
Here  the  factors  are  2hx^  Say,  c ;  and  the  l.  c.  m.  is  6ahcxy. 

Ex.  2.  Find  the  l.  c.  m.  of  2a^  (a  +  x),  4:ax  (a-  x),  Gx"^  (a  +  cc).  ' 
Here  the  l.  c.  m.  of  the  simple  factors  is  Ga^a;^,  that  of  the  com- 
pound factors  is  a^-a;*;  therefore  the  l.  c.  m.  required  is  12«V 


68  LEAST  COMIVION  MULTIPLE. 

Ex.  35. 
Find  the  l.  c.  m. 

1.  Of  ia'hc  and  (jah^c  ;  of  Ox^y  and  12xy^ ;  of  axy  and  a  (xy-y^) ; 

of  al)  +  ad  and  ah  -  ad. 

2.  Of  Sa\  lOa'h,  and  12a^Z)^ ;  of  a^  5a%  lO^j'J',  10a'6^  5«Z»*, 

and  i'' ;  of  9x^  Gaaj,  Sa'',  3GaJ^  3aa;^  SOa^a;,  and  2Aa\ 

3.  Of  2(a  +  &)  and  3(a'-&'^);   of  4(a^-a)  and  6  (a"  +  a) ;  of 

6  (ic^  +  ic?/),  8  (ojy  -  y^),  and  10  (ic^  -  ^^). 

4.  Of  4  (a^-a2»^),  12  (^&'  +  6=),  8  (a''  -  a'Z>)  ;  and  of  6  (x^'y  +  xy^\ 

9  {x^  -  xy'^),  4  (y^  +  icy^). 


YO.  Every  common  rriulti/ple  of  a  an^^  b  ^6*  a  multir 
fie  of  their  l.  c.  m. 

For  let  J/be  any  common  multij)le  of  ^  and  J,  and 
m  their  l.  c.  m.  ;  and  let  M  contain  m  (if  possible) 
T  times  with  remainder  5,  which  will  of  course  be  less 
than  the  divisor  m  ;  hence  we  should  have 

M=-  rm  +  5,  and,  therefore,  s  =  M-  rm  : 
but  since  a  and  5  measure  both  J/"  and  ?;?,  they  would 
also  (63)  measure  If-rm^  or  ,5;  i.  e.  5,  which  is  less 
than  m,  would  be  a  common  multiple  of  a  and  J, 
contrary  to  our  supposition  that  m  was  their  least 
common  multiple.  Hence  Jf  will  contain  m  with  no 
remainder,  and  will  therefore  be  a  niidtvple  of  m. 


CHAPTER    VI. 

FEACTIONS. 

Algebeaical  Fractions  are  for  the  most  part  pre- 
cisely similar  both  in  their  nature  and  treatment  to 
common  Arithmetical  Fractions.  We  shall  have, 
therefore,  to  repeat  much  of  what  has  been  said  in 
Arithmetic ;  but  the  Kules  which  were  there  shewn 
to  be  true  only  in  the  j^articular  examples  given,  will 
here,  by  the  use  of  letters,  which  stand  for  any  quan- 
tities, be  proved  to  be  true  in  all  cases, 

71.  A  Fraction  is  a  quantity  which  represents  a 
part  or  parts  of  an  unit  or  whole. 

It  consists  of  two  members,  the  numerator  and  de- 
norninator^  the  former  placed  over  the  latter  with  a  line 
between  them.  Now  we  have  already  agreed  (8)  that 
such  an  expression  shall  denote  that  the  upper  quan- 
tity is  divided  by  the  lower;  and,  in  accordance  with 
this,  it  will  be  seen  presently  that  a  fraction  does  also 
express  the  quotientof  the  num''  divided  by  the  den^ 

The  den""  shews  into  how  many  equal  parts  the  unit  is 
divided,  and  the  num'  the  number  taken  of  such  parts. 

Thus  y-  means  that  the  unit  is  divided  into  1  equal  parts,  a  of 
which  are  taken. 

Every  integral  quantity  may  be  considered  as  a 

fraction  whose  den  "■  is  1 ;  thus  a  is  - . 

Y2.  To  multi;plij  a  fraction  by  an  integer,  we  may 
either  multiply  the  num'or  divide  the  den' by  it; 
and,  conversely,  to  divide  a  fraction  by  any  integer,  we 
may  either  divide  the  luiiu''  or  multiply  the  den'  by  it. 


60  FRACTIONS. 

Thus  ■=■  X  X  =  -J-:  for  in  each  of  the  fractions  ^ ,  -7-  > 
00  00 

the  unit  is  divided  into  h  equal  parts,  and  x  times  as 
many  of  them  are  taken  in  the  latter  as  in  the  for- 
mer ;  hence  the  latter  fraction  is  x  times  tlie  former, 

^.    ^.  ax     a  1  v     •    -1  .       ax  ^        a 

that  IS  -=-=  -  X  X :  and,by  smiilar  reasoning,  - — -x=j , 

A^cain  -  ^  rx^'  =  — - ;  for  in  each  of  the  fractions  -7,  — - 
°       &  Ix'  Vox 

the  same  number  of  parts  is  taken,  but  each  of  the  parts 
in  the  latter  is  ~th  of  each  in  the  former,  since  the  unit 

X 

in  the  latter  case  is  divided  into  x  times  as  many  parts 
as  in  the  former ;  hence  the  latter  fraction  is  -tli  of  the 

X 

former,  that  is,  ^~—  ~j-^x\  and,  similarly,  ^—  x  a?  =  ^. 
'  'hx      b  '  '^^  Ix  b 

73.  If  any  quantity  be  both  multiplied  and  divided 

by  the  same  quantity,  its  value  will,  of  course,  remain 

unaltered.    Hence  if  the  num''  and  den^  of  a  fraction 

be  both  multiplied  or  divided  by  the  same  quantity, 

its  value  will  remain  unaltered. 

a      ax      a^        „  ,   a^h       a     ac        „ 

Ihus  -=--=—--  &c.  and  ----  =-=-—-  ~  &c, 
0      ox      ab  a^oG      c      c* 

74.  Since  c^  =  -  (71),  and,  therefore,  a  divided  by  b 

^---^b=  J  (72),  it  follows,  as  stated  in  (71),  that  a 
fraction  represents  the  quotient  of  the  num^  by  the  den'. 
In  factj  we  may  get  --  tli  of  a  units,  (or  a  -*-  h.)  by  taking  -  th 
part  of  each  of  the  a  units,  and  this  is  the  same  as  a  such  parts  of 
one  unit,  which  (71)  is  expressed  by  y. 

Hence  it  is  that,  in  Arithmetic.  I  of  £3  is  tlic  suiiic  as  ^'  of  £1.  ^.c. 


FRACTIONS.  61 

75.  To  reduce  an  integer  to  a  fraction  with  a  given 
denominator,  multiply  it  by  the  given  denominator, 
and  the  product  will  be  the  numerator  of  the  required 
fraction. 

Thus  a,  expressed  as  a  fraction  with  den'  oj,  is  — :  or,  with 

den'  0  -  c,  is  —^ . 

6-c 

The  truth  of  this  is  evident  from  (73). 

76.  The  signs  of  all  the  terms  in  both  thenum"^  and 
den'^of  a  fraction  may  be  changed  without  altering  its 

value :  thus  — ;r-  is  identical  with  — - — •. 

dax-x'  X  -"dax 

This  follows  also  from  (73),  as  the  process  is  equiva- 
lent to  that  of  multiplying  both  num""  and  den""  by-1. 

77.  To  reduce  a  fraction  to  its  lowest  terms,  divide 
the  numerator  and  denominator  by  their  g. cm. 

^x^y"^  o?x^if  axy 


Ex.1. 


a^xy  +  axy"^      axy  {a  +  y)      a  +  y 


^  ^  (y   a^  +  x^  _(a  +  x)  (a^  -ax  +  x-)  _  «'  -  ax  +  aj* 

a^  -X'  (a  +  x)  (a-x)        ~       a-x      ' 

^     ^    x^  +  4x  +  o      (aj  4-  3)  (a?  +  1)      x  +  1 
Ex.  3  V        /  V        /  _ 


Ex.4. 


x'^  +  5x  +  6      (x  +  Z)  (x  +  2)      X  +  2' 
x^  +  x^  +  Zx-6      (x  - 1)  (x^  +  2ic  +  5)     x"^  +  2x  +  5 


x^-4x+S  (.<;-l)0c-3)  aj-3      * 

Of  course,  the  student  should  consider  for  a  moment  whether 
he  cannot  obtain  the  g.  c.  m.  as  in  (58)  hy  mere  inspection. 

Ex.  36. 

Reduce  to  their  lowest  terms 
axy+xy^       cx+x'        llm'^  +  22mx      I4x^-7xy       5«'6-15a^i* 


axy     '     a^c  +  a\v'     S'6  (rrr-ix^) '     lOax-^iay'     20ah^+l0a-i'^' 

6x'^y-12xy^      2m^n-^2mu^      dHic^alPc^abc^      Vlx^y--2\xy''' 
al)c-\- %c-5c^  ae  +  hy  +  ay  +  he        acx'^  +  (ad-hc)  x-hd 


2ahdf-\-\mif~i{)cdj      of^2bx  +  2'>x-^hj  a^x'-b- 


62  FRACTIONS. 

.   aj'-l        a;*--a*  «•-&•        x^-h'^x  a^  -  ah  +  ax-  Ix 

ax+x     x^—a^x^  a^-V     x'^  +  2bx+b^  a^  +  ab  +  ax  +  Ix 

x'-ix  +  Z  x""  +  2x-S  g'-    al>-2b^ 

'^^2x  -  a'  x^  +  bx  +  fj  'a"-  Zab  +  21)'' 

Qa'^-lZax+Q^x''  (ja'^  +  lax-Zx'  ic^+iC-12 


^•*-«     T      «  -I  o     o  ^1  -I     -      •}  /»      a*  J.-J» 


5a;*  -  ISoj^y  +  ll^y^  -  Gy'*'  ""  a^oj  +  2a V  +  2aa;*  +  x^' 
a;*  +  3a;'^  -  4  a;^  -  3a;  +  2 

a;*  +  a^a;*  +  a*  -  3a^a;*  -  2ax'^  -  1 

a:*  +  ax^  -  a^x  -  a*'  '  4a^x^  -  2a'^x^  -  '6ax^  +  1' 


78.  If  the  niim''  be  of  lower  dimensions  than  the 
den'',  the  fraction  may  be  considered  in  the  light  of  a 
proj)er  fraction  in  Arithmetic  ;  if  greater,  in  that  of  an 
iinproper  fraction,  which  may  be  reduced  to  a  mixed 
fraction^  by  dividing  the  num*"  by  the  den^,  as  far  as 
the  division  is  j^ossible,  and  annexing  to  the  quotient 
the  remainder  and  divisor  in  the  form  of  a  fraction. 

Conversely,  a  onixed  fraction  may  be  reduced  to  an 
improper  fraction,  by  a  process  similar  to  that  em- 
ployed in  Arithmetic. 

_^     ,     3a;«  +  2a;  +  l     ^       ,.        41 
Ex.  1. =  3a;  - 10  +  j. 

a;+  4  a;  +  4 

Ex.  2.  a;^  +  a;  +  1  + 


x-\      x-V 
Ex.  37. 

1.  Reduce  to  mixed  fractions 

3a;V6a;4-5     a'^-ax^x''     2a;'' +  5      10^17«^+10a;'    16(3a;'  +  l) 
a;  +  4'        a  +  a;'     a;-3'  5a -a;  4a; -1* 

2.  Reduce  to  improper  fractions 

,  ^      3a;(3-a;)        ,    .        .  ,        Ca;»  a^-ay^y'' 

x^-^x ^-TT^,     a^-2aa;+4a;^ tt-,    a;-a  +  2/+ • ^-^. 

a;  -  2  '  a  +  2r  a;  +  a 


FRACTIONS.  63 


Shew  that 

3.  1  + ^r—. = i^— , ,  and 

2ah  2ab  ' 

a^  +  h^  -  c^  _  (a  -  h  +  c)  (h  -  a  +  c) 


2ab  2ab 

a      .w    .h^-c'X^      (a  +  h  +  c)  (a  +  h  —  c)  (a  +  c-h)  (b+c-a\ 


[       ^ 


79.  To  reduce  fractions  to  a  common  den"",  multiply 
the  num"^  of  each  fraction  by  all  the  den"  except  its 
own,  for  the  new  num'  corresponding  to  that  fraction, 
and  all  the  den'^  together  for  the  common  den^ 

The  truth  ofthis  rule  is  evident;  since,  the  numerator 
and  denominator  of  each  fraction  being  both  multiplied 
by  the  same  quantities,  viz.  the  denominators  of  the 
other  fractions,  its  value  will  not  be  altered,  though 
all  the  fractions  will  now  appear  wuth  the  same  de- 
nominator. 

Ex.  Reduce  - ,    -,     -,,  to  a  common  denominator. 

Z>'     c      d 

For  the  nunV  a  x  c  x  d  =  acd 

I  X  b  X  d  =  b^d  and  the  required  fractions  are 


c  X  b  X  c  =bc'^ 


acd     ¥d      be" 
bed)     bed)     bed' 


For  the  den'     b  x  c  ^  d  =  bed  ; 

80.  If,  however,  the  original  den''^  of  the  fractions 
have,  any  of  them,  common  factors,  this  process  will 
not  give  them  with  their  least  common  den"",  which,  as 
in  Arithmetic,  will  be  found  by  forming  the  l.  c.  m.  of 
the  given  den"  :  and  the  num'  corresponding  to  any  one 
of  the  given  fractions  will  be  obtained,  by  multiply- 
ing its  numerator  by  that  factor,  which  is  obtained  by 
dividing  the  l.  c.  m.  by  its  denominator. 

Ex.  Eeduce  ^7—,  -— — , to  a  common  denominator. 

Ibx    Gabxy     oacx 

Here  the  l.  c.  m.  of  the  denominators  being  Gabcxy,  the  fractions 

.     ,  ^a^cy  c^  2b'y 

Gabcxy^  Gabcxy'  Gabcxy' 


64  FRACTIONS. 


Ex.  38. 
Reduce  to  common  denominators, 

X     y      z       iB^        1/       s^  ,    2^^f/      33;*       4y^      5xy* 
^'  a'     h'    c'     2ab'    Tac'    4J^'    "3a^'     4a^'     6a6^'      W  ' 
x^  y^         a  +  X     a-x  Ax^  xy 


^'a'  +  b"-'     a'-b''    a-x'     a  +  x'    3(a  +  &'     6(a»-&^)' 

3.      1  ^  1 


81.  To  add  or  subtract  fractions,  reduce  them  to 

common  den",  and  add  or  subtract  the  num'"'  for  a 

,    new  num'',  retaining  a  common  den'. 

^     ^    x     y     z      hex  +  acy  +  ahz 

Ex.  1.  -  +  ^^  +  -  = ^ . 

a      0      c  aoc 

Ex.  2.    Add  :; 1    + 


Ans, 


1  +  X  +  x"^      1  -X  +  x" 
(l+x)(l-x  +  x"")  +  (1  - x)  (1  +  x  +  x') 


(1  +  a;  +  x'')  (1  -  ic  +  oj')  1  +  if' 


Ex.  3.  From  ^ -„  take 


Ans, 


1  +  X  +  x"^  1  -  X  +  x"^' 

(1  +  ic)  (1  -  ic  +  a;^)  -  (1  -  ic)  (1  +  cc  +  x^")  2x* 


(L  +  X  +  x'^)  (I  -  X  +  x'^)  1+x^+x* 

Ex.  4.  Find  the  value  of  2  +  ~ — ,-„ r. 

a^  -  b^      a  +  b 

2  (g^  -  b'')  +  (g'  +  b^)  -{a-  I)  {a  -  b)    ^a" -\-2ab -21^ 
Arts.  ^—^-^i  ________ 

Ex.  39. 

Find  the  vahie  of 

1     ^ _  (^ ~  ^)    iL      (^  "^  ^^^     ^^ ~ ^^     ^''^ ~l-e     \ba  -  Ac 
'    2b     2  (a  +  bf  2b  "^  Z~(cr^y       2~"  ^         ^       12 

p      ci*    ^  a  b  a  b      a-b         ab 


a  -b       ■   a  +  b      a-b'  a-b      a  +  b   a+  b     a^  -  b"^' 
a     (ad  -  be)  x   a^  +  b*     a-b   2.c^  -  2xy  +  y""      '  x 


•   -  -  —7 tV, -,-;.  ± 

4. 


c      c(e  +  dx)^  a^  -b^     a  +  V        x"^  -  xy  x-y' 

1  1  a        a~be-ah-c 


2(a-jc)      2Qi  +  x)      a"  +  a-^'     ab  ae  be  ' 


FRACTIONl*.  65 

6.   rr =    + 


2(a;-l)     2(a;+l)      cc^ '  '  2a+b     2a-b     4a^-h^' 

7  ^      (a''-'b'')x     a(a^-&V  J^ 1__      ^-1 

5          5'^  "    ■*■    b''{b  +  ax)  '  '  x"      (a;^  +  iy  "*"  ic'+l* 

9  ^yl  _  __^._  +  _5_  10. 1  -  ^^  +  ^'  .  -?^y 


a? -2/       ic-y     ii'+T/  ic+2/     or-y^      x^  +  y^ 

11.4,  ^Zfi^.       12.  2-?;:^!.5:i<.      13.  4- -f^^. 

a^      a(a+x)  x^  +  y     x'-y^               a^      a(a-x) 

14.  ^'^y       ^       x^y-x^  1-      ^           ^^            ^* 


82.  To  multiply  one  fraction  by  another,  multiply 
the  numerators  together  for  a  new  numerator,  and  the 
denominators  for  a  new  denominator. 

Suppose  that  we  have  to  multiply  -  by  -  : 

let  -  =a?,  j=y;  .',  a=:  hx,  c  =  chj,  and  ac  =  hdx?/; 

CiC 

hence,  (dividing  each  of  these  equals  by  hd)^  :~—'xy\ 

bd 

,    ,  a       c         ^ac      axe      product  of  num" 

but  w  =  Y  X  -  ,  and  ^,  =  -. — -  ~        ,         ^  , 

o      d  bd      bxd      product  01  den" 

whence  the  truth  of  the  rule  is  manifest. 

Similarly  we  may  proceed  for  any  number  of  fractions. 

a+b     a-b      3  _  Z{a+b)  {a-  b)      3  {0"  -  &^) 

Tld""  c-d''  2~  2{c  +  d)  {c  -d)  ^  2  (c^  -  d"") ' 

83.  To  divide  one  fraction  by  another,  invert  the 
divisor  and  proceed  as  in  Multiplication. 

Suppose  that  we  have  to  divide  -7  hy  , : 

let  ~z=x^  ~  =  y]  .*.  a  =  bx,  c=  dy ; 

,  7777        77  -3  ctd       hdx       X 

hence  ad  =  bdx.  be  =  bdi/,  and  :=—  =  7--  =  - ; 
'  •^'  be        bdy       y' 


66  FRACllONS. 

,    ^  a?  a      c         .  ad      a      d 

but  -=zx  -^7/  =:  y-f--^,  and  -—  =  -  X  -, 

y  -^        h      d^  ho       h       c' 

whence  the  truth  of  the  rule  is  manifest. 

2a  +  h  ^     c-cl    _{2a+U)  (2a  -  Zh)  _  4a"  -  96" 
c  +  d    '  2a-W~     {c-^d)(c-d)     "    c'-d^     • 
In  mult"  and  div"  of  fractions,  it  is  always  advisable,  before 
multiplying  out  the  factors  of  the  new  num '  and  den',  to  see  if 
some  of  them  do  not  exist  in  loth  the  num'  and  den',  in  which 
case  they  may  be  struck  out,  and  the  result  will  be  more  simple. 
^..acx_ac_ac 

'  bx      d      id      Id' 
p  5ax     xy  +  y"^  _  5a  (x  +  y)  _  5ax  +  5ay 

'6cy      x"^  -  xy      3c  (x  —  y)      Zc  (x  —  y)  ' 
^      o    4aa;       a"^  -  x^      he  +  hx     Ax  (a  +  x)     Aax  +  4x^ 
Zby        c^  -  X'      a^  -  ax      Zy  {c  -  x)      Zy  (c-x)' 
p       .    X'  +  xy     X*  —  y*      ic"  +  xy      (x  —  yy  _      x 

JilX.     4.  ■  -i-  -  r^  =  ■    X  -  -—    =  — -  -  , 

x  —  y       {X-  yy       x  —  y         x^-y^        x^  +  2/ 
The  student  should  leave  the  denominators  of  fractions  with  their 
factors  unmxdtijolied^  as  in  Ex.  2  and  3 ;  unless  they  happen  to 
combine  very  simply,  as  {a-^x)  (a+xy  into  (a+xy,  or  (a  +  x)  (a-x) 
into  a^  -  a;^.     The  convenience  of  this  will  be  found  in  practice. 

Ex.  40. 

Find  the  value  of 

_    2x     Zah      Zac        ax         a'-x^      a       /,     hx\       /_       a  \ 

1.    _  X   . X  -—-, X   ■ —-      --  X    [0+  —      X       1 . 

a        c         2b      (a-xy        ah    '     bx      \       a]       \       a  +  xj 

9,    i  -^l\(^     ^\     ^^'~^'  ^  ja-^xy   2a {x'^-yy x^ 

'   \      a  j  \x      ay    a^  +  x^      (a-x)'^  ex         ""  i^-y)  (x+yy' 

a"  +  2a&  ^  «&_-2&^  ^'^y  ^  {x-yy  I  4_  ^\  ^  aV+qZ>a;'  ^    ax 
*   a'  +  W  ""  a"'--4>'  ~x^y  ""  l^y^'  \^  ~  ic=  /        ax+l~  "^  a^^' 

4.  "-^?-:^^-|,  tiyl^-'--y^y\  (u^\^H,A] .  fi  j: 

a^-V    a  +  h    x'-y^  x-y         \      x)       \       x]      \     x^ 

^    I.      V\      (a     h\  a^-Za'^b^ZaW-b'      2ab-2h^     a^+ab 

V       '^y"*"U^^J'  ^"^^           '^~~J~"'^b' 

x*-b*          x''  +  bx  ic'-JV      ic*-2Ja;'+6V 


x^-2hx  +  b'^  '     x-h        x^  +  h^  x'^-bx  +  h'^ 


FRACTIONS.  67 

84.  A  complex  fraction,  ^.  e,  one  in  which  the  niim^,  or 
den^j  or  both,  are  fractions,  may  be  simplified  as  follows. 
^      X     2-x 

2  _  ^J^  _  2-x      l^  _  2-x 
Ex.  1.     4x    ~    4x    ~     2     ""  4x~  '  8x '' 

T 

Hence  observe  that,  when  a  complex  fraction  is  put  into  the  form 

of  a  t; ;—  ,  the  simple  expression  for  it  will  be  found  by  taking 

the  product  of  the  upper  and  lower  quantities,  or  extremes,  for  the 
num',  and  that  of  the  two  middle  ones,  or  means,  for  the  den' ; 
and  that  any  factor  may  be  struck  out  from  either  of  the  extremes,' 
if  it  be  struck  out  also  from  one  or  other  of  the  means. 


Ex.2. 

2x 
2x           I 
1  ~  3a;  -  I  ~ 

ex 

^x  -  r 

Ex.3. 

20 -aj 
5-lx         4           60  -  3a;  . 
ic  +  11     3a!  +  4     4  (3a;  +  4) 
3 

x+2 

x+2 

Ex.  4. 

(a!+2)  (2a;-a;=  +  3)       6  +  7a;-a;' 

(l-x)  (2a;-a!V3)+a;     3-3a;''*+a;^* 


3-a;       3a;  +  2A     2|  -  ja;      x-^ 
x'     X  +  2i'         31      '     fx  -  Ip     2 J  -  Ix 
2   ^-i(^^-^)      6a;  -  f  (3  4-  bx)'    2^-1  (x~2)      l|-|(a;+2) 
3  '  2^  '     |(^4-l)-4l'       tVC-c  +  1) 

o}  -vV^         25^        «  +  a;      r?-  -  a; 

2<x^     '    a  +  a;      <^/.  -  a;' 
o?  +  IP'      a  —  X      a  +  X 


68  FRACTIONS. 

85.  The  following  results  should  be  noticed. 

If  T  =  -^,  then 
h     a 

.      a      ^      c  1)      d  ^..      a      1)      1)      c  ah      .... 

1  -J-  -  =  1  -^  -„    or  -  =  --  (i),     Y  X  -  =  -  X  -    or    -  =  -,     00  ; 
l  cV         a     c  ^^^     h     c      c      cV  c      d     ^  ' 

a  ±1)      h      c  ±  d     d  a  ±b      c  ±  d  .  . 

hence  —7—  x  -  -  — — -  x  -,  or = (v), 

0         a         d         c  a  c  ^ 

^  a  +  h         h         c  +  d        d  a  +  h      c  +  d  ,  .. 

and  --.r-  ^  1  =  — j-  ^ n,   or    =  =  ,  (vi) : 

0         a  -0         d         c  -  d  a  —  0      c-  d 

and  any  of  these  last  may  be  inverted  by  (i),  ov  alternated  by  (ii) ; 

,  a  c         aa±ha  +  ha  —  I>o 

thus r  = i,     -  =  -— -  „ V  = ^J  &c. 

a  ±  0      c  ±  d     c      c  ±  d     c  +  d      c  -d 

So  that,  If  any  two  fractions  are  equals  we  may  com- 
bine  hy  Addition  or  Suitr action^  in  any  way^  the  nur)i^ 
and  den^  of  the  one ^  provided  that  we  do  the  same  with 
the  other, 

86.  The  above  results  may  be  yet  further  generalized. 

_       .^a      c    ,.        m      a      m      c        ma      mc 
For,  \1y=~t,  then  —  x   -  =  —  x  -,  or  — r-  =  — .\ 
^      0      d  n       0      n      d        710       nd 

and,  therefore,  by  what  has  been  above  she«rn, 

ma±n'b     mc±nd      .  ma±n'b      mc±nd       ,  ma±nh    me±nd 

.  whence •  = ,  and  = ; 

ma  nc     ^  a  c  pa  j>^ 

ma±nh      mc±nd   ma±nc     mh±nd   ma±nc      mh±nd 

so  also  Y—  =  —3 — , = — ,  = -z — , 

pb  pd  pa  P^  P^  P^ 

ma  +  nh     me  +  nd     ma  +  nb     ma  -  nb   „ 

ma  -  nb     7nc  +  nd)     mc  +  nd     mc  —  nd) 

.     .        .       ma  ±  nh      mc  ±  nd        .pa±ob     pc  ±  qd 

Again,  since = ,  and^^ ^  = --, 

a  c      '  a  c 

ma  ±  nb     mc±  nd   . 

2ja  ±  qb     pc  ±  qd) 
Hence  we  see  that  the  statement  of  (85)  is  true  of  any  mnltlpUa 
whatever  of  the  numerators  and  denominators  of  the  fractions. 

87.  Further,  rf  ^  =  ^-^  then^,-  =  ^„    -^,  =  ^,  &c.  ^-  =  ^-. 
Hence  the  previous  results  hold  with  a",  b'\c''^d'*^  instead  of «,  K  c,  d. 


FKACTIONS.  69 

For  let  T  =  ^  =  3  =  ^  >  t^i^Ji  a  =  dx,  c  =  (Zaj,  <?  =/a; ; 

,\a  +  c  +  e  =  hx  +  dx  +/x  =  (h  +  d  +  f)x'y  »\  x  ov  ■=-=  , — ^ : 

0     0  +  a  +j 
again,  ma  =  wJaj,  nc  =  ndx,  pe  =  pfx ; 

.    ,         7       ^^  J  ct     ma  +  nc  +  pe 

.%  ma  +  n(j  +  »g  =  (mo  +  nd  +  pj)  x,  and  a;  or  t  =  — ^j^ :; — ^--. 

0      mb  +  nd+pf 

c     ,     «••  _  c**  _  e**         fi'*  _  a"  +  c"  +  e*  _  TTia**  +  tic**  +  joe** 
boalso^-^-— ,  .-.  — -  j„  ^  ^.  ^^„  -  ^j„  ^  ^^n  ^^y-«. 

N.  B.  The  above  method  of  proof  will  evidently  serve,  whatever 
be  the  number  of  equal  fractions. 


89.  We  know  by  Division  that  the  fraction 

= =  1  +  X  +  X-  +  x^  +  &c.  +  x**~^  + ; 

1-X  I -X 

JC**  1 

so  that  :i will  be  the  difference  between  = —    and  the  first 

1-x  1-x 

n  terms  of  the  series :  and  this  difference,  if  ic  be  <  1,  becomes  less 

and  less  by  increasing  n^  that  is,  by  taking  more  terms  of  the 

series ;  whereas,  if  ic  be  >  1,  it  becomes  greater  and  greater.  Hence, 

when  a;  <  1,  the  fraction  = expresses  approximately,  and  with 

more  and  more  of  accuracy,  according  as  we  take  more  terms,  the 
value  of  the  series  \  +  x  +  x^  +  &c. ;  whereas,  when  x  >  1,  it  does 
not  at  all  express  the  value  of  the  series,  unless  we  take  account 

also  of  the  remainder  = ■ . 

1  -X 

Thus,  if  a;  =  I,  we  have  —-j  or2  =  l  +  |  +  |^  +  J+  &c.,  the  sum 

of  which  series  approaches  more  and  more  nearly  to  2  as  its 
Limitj  without  ever  actually  reaching  it.     But  if  a;  =  2,  we  have 

=— -  OP -1  =  1  +  2  +  4  +  8  +  &c.,  the  sum  of  which  series  departs 

1  — ii 

more  and  more  from  -1 :  the  error,  however,  will  be  corrected,  if 
we  introduce  the  remainder  at  any  step ;  thus 

1+2  +  4+ ~2  =  7-8  =  -l. 


70  FKACTIONS. 

In  all  such  cases  we  may  consider  the  sign  =  as  expressing,  not 
the  actual  equality  of  the  two  quantities,  but  merely  that  the 
fraction  can  be  made  to  assume  the  form  of  the  serieSj  and  there- 
fore may  be  used  as  an  abridgment  for  it. 

90.  If  a;  =  1  in  the  above,  'then  = — =   =  1  +  1  +  &c.,  that   is, 

X  =  an  infinite  number  of  units,  which  is,  of  course,  an  infinitely 

great  quantity,  and  is  denoted  by  oo   (read  infinity). 

The  meaning  of  this  result  may  be  thus  explained.    If  ic  =  1 

'cery  nearly^  so  that  \-x  is   'cery  small,  then  j — ^  will  be,  of 

course,  very  great,  and  may  be  made  as  great  as  we  please  by 
still  farther  diminishing  1-x,  that  is,  by  taking  x  still   more 

nearly  =  1.    T/^hen,  therefore,  we  write  ^  =  oo  ,  we  are  not  to 

suppose  the  denominator  actually  zero  (in  which  case  the  division 
by  which  we  obtained  the  scries  would  be  absurd),  but  only  a 
very  small  quantity ;  and  by  using  the  sign  <x) ,  we  mean  that 

there  is  no  Limit  to  the  magnitude  which  the  fraction  zr may 

be  made  to  attain,  by  sufiSciently  diminishing  the  denominator. 

In  the  same  sense,  we  may  say  that  ur  =  oo  ,  where  a  represents 

any  finite  quantity  whatever. 


CHAPTEE   VII. 

SIMPLE  EQUATIONS   CONTINUED. 

91.  The  following  equations,  involving  algebraical 
fractions,  may  now  easily  be  solved,  by  help  of  the 
preceding  chapter,  after  the  methods  in  Ex.  18, 19. 

Ex.  42. 

-            hx     ax-  ¥ 
1.  a = . 

a  c 

.    ax  +  h     a  _  ex  +  d 
c         b~~      e      ' 
^    a        h         -     ^_  „   ax     ex  1  z^i        n 

^-  J^-S5  =''-*•  7.  ^.  -^  =  ^.  .  yifh-cx). 

8.  \  \iaO-+x)-iia,-x)]  =  J  {3a(l-a')  --'/(a  + »)}. 
.2a:  +  3     Ax     1     Ore +  2     «  +  l 


a 

1) 

d 

X 

X 

ii. 

+  — 

=  0. 

3. 

~  + 

=  (?. 

x 

c, 

e 

a 

b 

5. 

aid'' 

+  x') 

ac 

+ 

ax 

a 

Ix 

d 

10.1-^ 


4      "^    3       x'^      3  6 

X 


92.  Complex  fractions  in  an  equation  should  first  be  reduced  by 
(84)  ;  and  if,  in  any  case,  the  denominators  contain  both  simple 
and  compound  factors,  it  is  best  to  get  rid  of  the  simple  factors 
first,  and  then  of  each  compound  factor  in  turn,  observing  to  sim* 
plify  as  much  as  possible  after  each  multiplication. 

X  +  1         3a; +  2  x  +  I 

Here,  first  simplifying  the  complex  fractions,  we  get 
Y5-a;        80a;  +  21      ^       23 


3  (a;  +  1)      5  (3a;  +  2)  x  +  1' 

^,              ,^.  ,  .       ^     _    375 -5a;     240a;  +  63  ^^       345 

then,  multiplymg  by  15,  = —  +  — ts pr-  =  75  +  =■ : 

'            i-^oj       ^     X  +  1           3a; +  2  a;  +  l' 

li.  ,-           -.    o^r     r        240a;' +  303a!  +  63  ^,       ^^     „,, 

.•.,  mult,  by  a;  +  1,  375  -5a;  + r. ?i =  /oa;  +  75  +  345; 

'  "^  '  3a;  +  2 


240a;' +  303a; +63 
3a; +  2 
240a;'  +  303a;  +  63  =  240a;'  +  295a;  +  90,  and  8x  =  27,  or  ic  =  3 


simplifying, ^ ^ =75a;+5a;+75+345-375=80a;+455 


72  SIMPLE   EQUATIONS. 

Ex.  43. 

-    aj  +  f  ^  25  _    4       5x  rt     2i?;      1  - 1^  _  a;  - 1     x 

8j;  +  5      7a;  -  3  _  4a;  +  6  2  (4a;  +  3)        3 

""14"  ""  6iT2  ~      7  ~'  ~^T3~  ■"  Sn  ~  ^* 

era;              5a;          -  r^~^_^      ^~^ 

*  5(a;  +  c)      a  (a;  +  c)  ~    *  '  x  +  2~2     2a;- 1 ' 

6a;  +  ^_  3a;- 5  a;-i(a;-l)      31  _  3-|(a;-2) 

4x^~  2x-a'  •   ■       3           "^SG"           5 

9.„4:i^._L_^=li.  10.       1  1  1 


J 


'3(3-a;)      2(l-a;)  ab-ax     hc-bx     ac-ax' 

(2a;  +  3)  a;      1_        .,  ^^     2a; +  a  3a;-fl^_ 

^^•~2^"Tr"  ■*■  35  "''■'•  -^  •  3"^^)  ""  2"(^7^  -  ^^■ 

iq    ii'>.     .  n   .  rV    .  l"l^^2f-f^V(a;-l) 
Id.  1^  {oa;  -  f  (1  +  a;;[  +  ~p-  = ^: 

-,       X         a  +  X     2a -h  ^^    x  +  4       _,       3a; +  8 

14. = ^ — .  15.  ^ ^  +  If  =  o — ^  • 

« +  a;        X  2x  6x  +  b        "^      2a;  +  3 

16.  ^V  (11a;  - 13)  +  i  (19a;  +  3)  - 1  (^^  " 25^  =  281  -^  (17^  +  4). 
10a;  +  17      12.r+2  _  5a;-4  a; +11      lO-a;  _  4-|a;      1 

"     18  i3a;-16~     9    *  3  "       3|~~"Tr"TI* 

19.  |(a;-lt|-)-TV(2-6a;)=a;-^V|5a;- 1(10-3^)}. 

..   6a; +  13       3a;  +  5      2a;  oi    ^-''     2a;- 15  1 


15         5a!-25      5'  '  a;  +  7      2x-6      2(a!  +  7)* 

„„  132x  H- 1     8a!  +  5  „,  7a!  +  l      35     a'+4 

24.  ^tl 1^=_J_.25   _1L_.-A___J_ 

6«  +  17     3x-10     l-2«  12a;  + 11     C«  +  5     42!  +  7 

26.  ias  -  H2^-3)-H3^-l)  _  3     x- -  jx  ^  2 


i(a!-l)  2"      3x-2     • 

27.  A  (7a;+Ci)  +  Jj  |ll«-i  (a;-li)}  =  J  (3a;+l)  +^  j43a>-i  (S-Sa')}. 

'^'*-13-2x'^^'^       2r^-*T5 3 • 

29.  4a!-i(*-2)-[2x-aa!-J,jl6-i(«  +  4)()]  =  |(x+2). 
g,6-Sa!       7 -2a!'.  _  1  +  3a; _  2a; -2j^     _1 
15        14(x-l)~     21  C       *  105" 


SIMPLE   EQUATIONS.  73 

93.  Tlie  toUowing  are  additional  Problems  in  Sim- 
ple Equations,  presenting  somewhat  more  of  difficulty 
than  those  given  under  (41). 

Ex.  1.  A  fish  was  caught  whose  tail  weighed  9  lbs;  his  head 
weighed  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighed  as  much  as  his  head  and  tail.     What  did  the  fish  weigh? 

It  is  sometimes  convenient  to  take  x  to  represent,  not  the  quan- 
tity actually  demanded  in  the  question,  but  some  other  unknown 
quantity  on  which  this  one  depends.  It  is  only  experience,  how- 
ever, and  practice  which  can  suggest  these  cases ;  but  this  ex- 
ample is  one  of  them. 

Let  X  =  weight  of  body ; 

.'.  9  +  |.r  =  weight  of  tail  +  \  body  =  weight  of  head; 
but  the  body  weighs  as  much  as  head  and  tail ; 

.*.  a;  =  (9  +  ^  x)  +  9,  whence  a;  =  36,  weight  of  bod}^ ; 
.'.  9  +  ^ic  ==  27,  weight  of  head  ; 
and  the  whole  fish  weighed  27  +  3G  +  9  =  72  lbs. 

Ex.  2.  A  gamester  at  one  sitting  lost  J  of  his  money,  and  then 
won  IO5  ;  at  a  second  he  lost^  of  the  remainder,  and  then  won  Zs ; 
and  now  he  has  3  guineas  left.  How  much  money  had  he  at  first? 
Let  X  =  number  of  shillings  he  had  at  first ; 
having  lost  4-  of  it,  he  had  |  of  it,  or  |  a;  remaining; 
he  then  won  10s,  and  had,  therefore,  |  a;  +  10  in  hand ; 
i)sing  \  of  this,  he  had  f  of  it  remaining,  that  is,  f  (|  a;  +  10) ; 
and  he  then  wins  3s,  and  so  has  f  (|  ^  +  10)  +  3  shillings, 
which,  by  the  question,  is  equal  to  3  guineas,  or  63s; 
hence  f  (|  a;  +  10)  +  3  =  63,  w^hence  x  =  100s  =  £5. 
Ex.  3.  Find  a  number  such  that  if  %  of  it  be  subtracted  from  20, 
and  y\  of  the  remainder  from  ^  of  the  original  number,  12  times 
the  second  remainder  shall  be  half  the  original  number. 
I '  Let  X  =  the  number  ; 

\\  20-|  X  =lst  remainder,  and  5-  ^  -  fi  (20  -  i  a-)  =  2nd  remainder ; 
.*.  12  {\x-  y\  (20  -^x)}  =1  re,  by  the  question  ;  whence x  =  24. 
Ex.  4.  A  certain  number  consists  of  two  digits  whose  difference 
is  3  ;  and,  if  the  digits  be  inverted,  the  number  so  formed  will  be 
f  of  the  former  :  find  the  original  number. 
4 


k 


74  SIMPLE   EQUATIONS. 

Let  x  =  lesser  digit,  and  .'.  oj  +  3  =  the  greater:  then,  since  the 
value  of  a  n°  of  two  digits  =  ten-times  the  first  digit  +  the  second 
digit  (thus  67  =  10  X  G  +  7),  the  n"  in  question  =  10  (a;  +  3)  +  cc ; 
similarly,  the  n"  formed  by  the  same  digits  inverted  =  10j;+  (x+Z)  ', 
hence,  by  question.  10a;  +  (a;  +  3)  =  -f  { 10  (x  +  3)  +a;  J-,  whence  aj  =  3, 
a;  +  3  =  6,  and  the  n"  required  is  63. 

Ex.  5.  A  can  do  a  piece  of  work  in  10  days ;  but  after  he  has 
been  upon  it  4  days,  B  is  sent  to  help  him,  and  they  finish  it 
together  in  2  days.    In  what  time  would  B  have  done  the  whole  ? 

Let  a;  =  n"  of  days  B  would  have  taken,  and  TT  denote  the  work : 

W   W 

,\  — r,  — ,  are  the  portions  of  the  work,  which  A,  B  would  do  in 
10     a;' 

4Tr 

one  day ;  hence  in  4  days,  A  does  -y^-,  and  in  2  days,  A   and  B 

,^     ^    2W     2W       4Tr     2W     2W     ^       ^ 

tofiretherdo  -^t,-  + :  .\  -ttt  +  -^-n  + =  f^'y  whence  x  =  o. 

^  10         a;  10        10        a; 

It  is  plain  that  in  the  above,  we  might  have  omitted  W  al- 
together, or  taken  U7iiti/   to  represent  the  work,   as   follows: 

A,  i?  do  y^,  -  of  the  work  respectively  in  one  day,  and  therefore, 

4       2      2 
reasoning  just  as  before,  tk  +  th  +  ~  =  ^^c  whole  work  =  1. 

[In  all  such  questions  the  student  should  notice  that,  if  a  person 

does  -  ths  of  any  work  in  1  day,  he  will  do  -  th  of  it  in  —  th  of  a 

n  n  m 

day,  and  therefore  the  wJiole  work  in  —  days. 

Thus  if  he  does  ^  in  one  day,  he  will  do  |  in  ^  of  a  day,  and 
.•.  ^  or  the  wJiole  in  5  =  2^  days]. 

Ex.  6.  A  cistern  can  be  filled  in  half-an-hour  by  a  pipe  A^  and 
emptied  in  20'  by  another  pipe  B :  after  A  has  been  opened  20', 
B  is  also  opened  for  12',  when  A  is  closed,  and  B  remains  open 
for  5'  more,  and  now  there  are  13  gallons  in  the  cistern :  how 
much  would  it  contain  when  full  1 

Let  X  =  number  of  gallons  that  would  fill  the  cistern :  then,  in 
1',  A  brings  in  75^5^3;  gals.,  and  B  carries  out  -^x  gals. ;  but  A  h 
opened  altogether  for  32',  and  B  for  17' ;  .*.  f|aj  -  \lx  =  1^ 
whence  a;  =  60  gals. 


SIMPLE   EQUATIONS.  75 

Ex.  7.  Find  the  time  between  two  and  three  o'clock,  at  which 
the  hour  and  minute-hand  of  a  watch  are  exactly  opposite  each 
other. 

Let  X  =  number  of  minutes  advanced  by  the  Jiour-hand  since  two 
o'clock  :  then  12ic  =  number  of  minutes  advanced  by  the  minute- 
hand,  since  it  ti-avels  GO'  while  the  other  travels  5' ;  but.  by  ques- 
tion, the  minute-hand  will  have  advanced  (10+a;)+30=a;+40  min. ; 
.-.  12ic  =  ic  +  40,  whence  x  =  Z^j^  and  the  time  is  2h  43yy'. 

Ex.  8.  There  are  two  bars  of  metal,  the  first  containing  14  oz. 
of  silver  and  6  of  tin,  the  second  containing  8  of  silver  and  12  of 
tin ;  how  much  must  be  taken  from  each  to  form  a^bar  of  20  oz. 
containing  equal  weights  of  silver  and  tin  ? 

Let  ic  =  n"  of  oz.  to  be  taken  from  first  bar,  20-0?  from  second  ; 
now  ^J  of  the  first  bar,  and  therefore  of  evevi/  oz.  of  it,  is  silver  ; 

and,  similarly.  2^  of  every  oz.  of  the  second  bar  is  silver ; 
and  there  are  to  be  altogether  10  oz.  of  silver  in  the  compound ; 
...  |4^  +  _Cg.  (20  -ir)  =  10,  whence  x  =  6|,  and  20  -  a;  =  13  J-. 

Ex.  44. 

1.  The  stones  which  pave  a  square  court  would  just  cover 
a  rectangular  area,  whose  length  is  six  yards  longer,  and  breadth 
four  yards  shorter,  than  the  side  of  the  square  :  find  the  area  of 
the  court. 

2.  Out  of  a  cask  of  wine,  of  which  a  fifth  part  had  leaked  away, 
10  gallons  were  drawn,  and  then  it  was  two-thirds  full :  how  much 
did  it  hold  ? 

3.  A  person  bought  a  chaise,  horse,  and  harness  for  £G0 ;  the 
horse  cost  twice  as  much  as  the  harness,  and  the  chaise  half  as 
much  again  as  the  horse  and  harness  :  what  did  he  give  for  each  ? 

4.  The  value  of  50  coins,  consisting  of  half-guineas  and  half-, 
crowns,  is  £16  55 :  how  many  are  there  of  each? 

5.  A^  after  spending  £10  less  than  a  third  of  his  ycarlv  income, 
found  that  he  had  £45  more  than  half  of  it  remaining  :  what  was 
his  income  1 

6.  A  boy,  selling  oranges,  sells  half  his  stock  and  one  more  to 
-4,  half  of  what  remains  and  two  more  to  i?,  and  three  that  still 
remain  to  C:  how  many  had  he  at  first  ? 


k 


76  SIMPLE   EQUATIONS. 

7.  In  a  garrison  of  2744  men,  there  arc  two  cavalry  soldiers  to 
twenty-five  infantry,  and  half  as  many  artillery  as  cavalry  :  find 
the  numbers  of  each. 

8.  A  person  dies  worth  £13^000:  some  of  this  he  leaves  to  a 
Charity,  and  twelve  times  as  much  to  his  eldest  son,  whose  share 
is  half  as  much  again  as  that  of  each  of  his  two  brothers,  and 
two-thirds  as^uch  again  as  that  of  each  of  his  five  sisters:  find 
the  amount  of  the  bequest  to  the  Charity. 

9.  A  farm  of  270  acres  is  divided  among  A,  B,  C:  A  has  7 
acres  to  11  of  i>,  and  G  has  half  as  much  again  as  A  and  B 
together :  find  the  shares. 

10.  Divide  150  into  two  parts,  such  that  if  one  be  divided  by 
23  and  the  other  by  27,  the  sum  of  the  two  quotients  may  be  G. 

11.  ^  had  I85  in  his  purse,  and  B,  when  he  had  paid  A  two- 
thirds  of  his  money,  found  that  he  had  now  remaining  two-fifths 
of  the  sum  which  A  now  had :  what  had  ^  at  first  ? 

12.  The  first  digit  of  a  certain  number  exceeds  the  second  by  4, 
and  when  the  number  is  divided  by  the  sum  of  the  digits,  tho 
quotient  is  7 :  find  it. 

13.  The  length  of  a  floor  exceeds  the  breadth  b}^  4  ft. ;  if  each 
had  been  increased  by  a  foot,  the  area  of  the  room  would  have 
been  increased  by  27  sq.  ft. :  find  its  original  dimensions. 

14.  A  met  two  beggars,  B  and  (?,,  and  having  a  certain  sum  in 
his  pocket,  gave  2^^  of  it  to^,  and  f  of  the  remainder  to  C:  A 
had  now  20^  left ;  what  had  he  at  first  ? 

15.  In  a  mixture  of  copper,  lead,  and  tin,  the  copper  was  5  lb 
less  than  half  the  whole  quantity,  and  the  lead  and  tin  each  5  lb 
more  than  a  third  of  the  remainder  :  find  the  respective  quantities. 

16.  A  sum  of  money  was  left  for  the  poor  widows  of  a  parish, 
and  it  was  found  that,  if  each  received  4«  6d,  there  would  be  Is 
over ;  whereas,  if  each  received  55,  there  would  be  10s  short ;  how 
many  widows  were  there  ?  and  what  was  the  sum  left  ? 

17.  A  horse  was  sold  at  a  loss  for  40  guineas;  but,  if  it  had 
been  sold  for  50  guineas,  the  gain  would  have  been  three-fourths 
of  the  former  loss :  find  its  real  value. 

18.  A  can  do  a  piece  of  work  in  10  days,  which  B  can  do  in  8 : 
after  A  has  been  at  w^ork  upon  it  3  days,  B  comes  to  help  him  ; 
in  what  time  will  they  finish  it  ? 

10.  There  is  a  number  of  two  digits,  whose  difference  is  2,  and, 


SIMPLE   EQUATIONS.  77 

if  it  be  diminished  by  half  as  much  again  as  the  sum  of  the  digits, 
the  digits  will  be  inverted  :  find  it. 

20.  A  and  B  have  the  same  income  :  A  lays  by  a  fifth  of  his : 
but  i?j  by  spending  annually  £80  more  than  A,  at  the  end  of 
4  years  finds  himself  £220  in  debt.     What  was  their  income  V 

21.  A  number  of  troops  being  formed  into  a  solid  square,  it  was 
found  there  were  GO  over;  but,  when  formed  into  a  column  with 
6  men  more  in  front  than  before  and  3  less  in  depth,  there  was 
Just  one  man  wanting  to  complete  it.     Find  the  number. 

22.  A  person  has  travelled  altogether  3036  miles,  of  which  he 
has  gone  seven  miles  by  water  to  four  on  foot,  and  five  by  water 
to  two  on  horseback  :  how  many  did  he  travel  each  way  ? 

23.  A  and  B  can  reap  a  field  together  in  7  days,  which  A  alone 
could  reap  in  10  days  :  in  what  time  could  JB  alone  reap  it  ? 

24.  A  cistern  can  be  filled  in  15'  by  two  pipes,  A  and  J3,  nin- 
ning  together :  after  A  has  been  running  by  itself  for  5',  B  is  also 
turned  on,  and  the  cistern  is  filled  in  13'  more :  in  what  time 
would  it  be  filled  by  each  pipe  separately  ? 

25.  What  is  the  first  hour  after  6  o'clock,  at  which  the  two 
hands  of  a  watch  are  (i)  directly  opposite,  and  (ii)  at  right  angles, 
to  each  other  ? 

20.  A  person  played  twenty  games  at  chess  for  a  wager  of  3^  to 
2^,  and  upon  the  whole  he  gained  5s :  how  many  games  did  he  win  ? 

27.  I  wish  to  enclose  a  piece  of  ground  with  palisades;  and  find 
that,  if  I  set  them  a  foot  asunder,  I  shall  have  too  few  by  150, 
whereas,  if  I  set  them  a  yard  asunder,  I  shall  have  too  many  by 
70  :  what  is  the  circuit  of  the  piece  of  ground  ? 

28.  A  and  B  began  to  pay  their  debts :  ^'s  money  was  at  first 
f  of  i?'s;  but  after  A  had  paid  £1  less  than  f  of  his  money,  and  B 
had  paid  £1  more  than  J  of  his,  it  was  found  that  B  had  only  half 
as  much  as  A  had  left.    What  sum  had  each  at  first  ? 

20.  A  can  build  a  wall  in  8  days,  which  A  and  B  can  do 
together  in  5  days :  how  long  would  B  take  to  do  it  alone  ?  and 
how  long  after  B  has  begun  should  A  begin,  so  that,  finishing  it 
together,  they  may  each  have  built  half  the  wall  ? 

30.  A  person  wishing  to  sell  a  watch  by  lottery,  charges  Cs  each 
foi:  the  tickets,  by  w^hich  he  gains  £4  ;  whereas,  if  he  had  made  a 
third  as  many  tickets  again  and  charged  5^  each,  he  would  have 
gained  as  many  shillings  as  he  had  sold  tickets :  what  was  the 
value  of  the  watch  ? 


78  SIMPLE  EQUATIONS. 

31.  A  mass  of  copper  and  tin  weighs  80  lbs,  and  for  every  7  lbs  of 
copper  there  arc  3  lbs  of  tin  :  how  much  copper  must  be  added  to 
the  mass,  that  for  every  11  lbs  of  copper  there  may  be  4  lbs  of  tin  ? 

32.  A  does  f  of  a  piece  of  work  in  10  days,  when  B  comes  to 
lielp  him,  and  they  take  three  days  more  to  finish  it :  in  what  time 
would  they  have  done  the  whole,  each  separately,  or  both  together  ? 

33.  A  cistern  can  be  filled  by  two  pipes,  A  and  B,  in  24'  and 
30'  respectively,  and  emptied  by  a  third  C  in  20' :  in  what  time 
would  it  be  filled,  if  all  three  were  running  together  ? 

34.  A  and  B  were  employed  together  for  50  days,  each  at 

a  day,  during  which  time  A^  by  spending  6d  a  day  less  than  B, 
had  saved  three  times  as  much  as  B,  and  2|  days'  pay  besides  ; 
what  did  each  spend  per  day  ? 

35.  Divide  £149  among  A^  B,  C,  i>,  so  that  A  may  have  half 
as  much  again  as  B,  and  a  third  as  much  again  as  B  and  G  to- 
gether ;  and  I)  a  fourth  as  much  again  as  A  and  C  together. 

3G.  There  are  two  silver  cups  and  one  cover  for  both.  The  first 
weighs  12  oz,  andj  with  the  cover,  weighs  twice  as  much  as  the 
other  cup  without  it ;  but  the  second  with  the  cover  weighs  a 
third  as  much  again  as  the  first  without  it.  Find  the  weight  of 
the  cover. 

37.  A  man  could  reap  a  field  by  himself  in  20  hrs,  but,  with  his 
son's  help  for  6  hrs,  he  could  do  it  in  IG  hrs:  how  long  would  the 
son  be  in  reaping  the  field  by  himself? 

38.  A  horsckeeper,  not  having  room  in  his  stables  for  8  of  his 
horses,  built  so  as  to  increase  his  accommodation  by  one  half,  and 
now  has  room  for  8  more  than  his  whole  number :  how  many 
horses  had  he  ? 

39.  A  grocer  bought  tea  at  6s  Gd  per  lb,  and  a  third  as  many 
lbs  again  of  coflee  at  2s  Gd  per  lb ;  he  sold  the  tea  at  8.9,  and  the 
coffee  at  2s  Zd,  and  so  gained  five  guineas  by  the  bargain ;  how 
many  lbs  of  each  did  he  buy  ? 

40.  Find  a  number  of  three  digits,  each  greater  by  unity  than 
that  which  follows  it,  so  that  its  excess  above  one-fourth  of  the 
number  formed  by  inverting  the  digits  shall  be  36  times  the  sum 
of  the  digits. 

41.  A  man  and  his  wife  could  drink  a  cask  of  beer  in  20  days, 
the  man  drinking  half  as  much  again  as  his  wife;  but,  ^|  of  a 
gallon  having  leaked  away,  they  found  that  it  only  lasted  them 


SIMPLE   EQUATIONS.  79 

together  for  18  days,  and  the  wife  herself  for  two  days  longer: 
how  much  did  it  contain  when  full  ? 

42.  A  and  B  have  each  a  sum  of  money  given  them,  which  will 
support  their  families  for  10  and  12  days  respectively  ;  but  J.'s 
money  would  support  JB^s  family  for  15  days,  and  i>'s  money  would 
support  ^'s  family  for  7  days,  -with  2s  Gd  over:  what  were  the  sums  ? 

43.  A  person  being  asked  how  many  ducks  and  geese  he  had  in 
his  yard  said,  If  I  had  8  more  of  each,  I  should  have  8  ducks  for  7 
geese,  and  if  I  had  8  less  of  each,  I  should  have  7  ducks  for  6 
geese  :  how  many  had  he  of  each  ? 

44.  A  man,  woman,  and  child  could  reap  a  field  in  30  hrs,  the 
man  doing  half  as  much  again  as  the  woman,  and  the  woman  two- 
thirds  as  much  again  as  the  child  :  how  many  hours  would  they 
each  take  to  do  it  separately  ? 

45.  If  19  lbs  of  gold  weigh  18  lbs  in  water,  and  10  lbs  of  silver 
weigh  9  lbs  in  water,  find  the  quantity  of  gold  and  silver  in  a  mass 
of  gold  and  silver,  weighing  106  lbs  in  air  and  99  lbs  in  water. 

4G.  From  each  of  a  number  of  foreign  gold  coins  a  person  filed 
a  fifth  part,  and  had  passed  two-thirds  of  them,  gaining  thereby 
355,  w^hen  the  rest  were  seized  as  light  coin,  except  one  with 
which  the  man  decamped,  having  lost  upon  the  whole  IfStf  as  much,  'fiS*^ 
as  he  had  gained  before  :  how  many  coins  were  there '^first  ? 

47.  A  and  B  start  to  run  a  race :  at  the  end  of  5',  when  A  has 
run  900  yards  and  has  outstripped  B  by  75  yards,  he  falls ;  but, 
though  he  loses  ground  by  the  accident,  and  for  the  rest  of  the 
course  makes  20  yards  a  minute  less  than  before,  he  comes  in  only 
half-a-minute  behind  B,     Ilovr  long  did  the  race  last  ? 

48.  A  and  B  can  reap  a  field  together  in  12  hrs,  A  and  C  in 
IG  hrs,  and  A  by  himself  in  20  hrs :  in  what  time  could  (i)  B 
and  0  together,  (ii)  A,  By  and  0,  together,  reap  it  ? 

49.  Fifteen  guineas  should  w^eigh  4  oz :  but  a  parcel  of  light 
gofd,  having  been  weighed  and  counted,  was  found  to  contain  9 
more  guineas  than  was  supposed  from  the  weight,  and  it  appeared 
that  21  of  these  coins  weighed  the  same  as  20  true  guineas :  how 
many  were  there  altogether '? 

50.  A,  B,  0  travel  from  the  same  place  at  the  rate  of  4,  5,  and 
6  miles  an  hour  respectively,  and  B  starts  two  hours  after  A : 
how  long  after  B  must  0  start,  in  order  that  they  may  both 
overtake  J.  at  the  same  moment  ? 


80  SIMPLE    EQUATIONS. 

Simultaneous  Equations  of  one  Dimension. 

94.  If  one  equation  contain  t%m  unknown  qnanti* 
ties,  there  are  an  infinite  number  of  pairs  of  values  of 
these  by  which  it  may  be  satisfied. 

Thus  in  ic  =  10  -  27/,  if  we  give  amj  xalue  to  t/,  we  shall  get  a 
corresponding  value  for  x,  by  which  pair  of  values  the  equation 
will  of  course  be  satisfied ;  if,  for  example,  we  take  7/  =  1,  we  shall 
getaj  =  10-2  =  8;  ify  =  2,  aj=G;  if2/  =  3,  aj  =  4;  &c. 

One  equation  then  between  two  unknown  quantities 
admits  of  an  infinite  number  of  solutions  ;  but  if  we 
have  as  many  diflferent  equations,  as  there  are  quan- 
tities, the  number  of  solutions  wnll  be  limited. 

Thus,  while  each  of  the  equations  x  =  10-2y^Ax  ■¥  4  =  Sy, 
separately  considered,  is  satisfied  by  an  infinite  number  of  pairs 
of  values  of  x  and  t/,  there  will  only  be  found  one  pair  common  to 
both,  viz.  a;  =  2,  y  =  4,  which  are  therefore  the  roots  of  the  pair 
of  equations,"  a;  =  10-2?/,  and  Ax  +  4:=  Zy. 

Equations  of  this  kind,  which  are  to  be  satisfied  by 
the  same  pair  or  pairs  of  values  of  x  and  y,  are  called 
shmdtaneous  equation s. 

If  there  be  three  unknowns,  there  must  be  three  equa- 
tions, and  so  on:  and  moreover,  these  equations  must 
all  be  different  from  one  another ;  i.  e,  must  all  express 
different  relations  between  the  unknown  quantities. 

Thus,  if  we  had  the  equation  a?  --  10  -  2?/,  it  would  be  of  no  use 
to  join  with  it  the  equation  2x  =  20-4?/  (which  is  obtained  by 
merely  doubling  it),  or  any  other,  derived,  like  this,  immediately 
from  the  former ;  since  this  expresses  no  new  relation  between 
X  and  7/,  but  repeats  in  anotlier  form  the  same  as  before.        • 

It  may  be  observed,  that  if  any  two  or  more  equa- 
tions be  given,  any  equations  formed  by  adding  or 
subtracting  any  multiples  of  these  equations,  will  be 
also  true^  though  expressing,  in  reality,  no  new  rela- 
tions between  the  quantities. 

Thus  if  ic  +  3?/  +  42  =  9,  and  2>x-2y  ^  Viz  =  25  j  then,  subtract- 
ing the  second  from  three  times  the  first,  we  have  \\y  -  62  =  2. 


SIMPLE   EQUATIONS.  81 

95.  There  are  generally  given  three  methods  for 
solving  smiultaneous  equations  of  two  unknowns;  but 
the  object  aimed  at  is  the  same  in  each,  viz.  to  com- 
bine the  two  equations  in  such  a  manner  as  to  expel, 
or,  as  the  phrase  is,  eliminate  from  the  result  one 
quantity,  and  so  get  an  equation  oione  unknown  only. 

dQ,  First  inethocl. — Multiply,  when  possible,  one 
equation  by  some  number,  that  may  make  the  coeff.  of 
X  or  y  in  it  the  same  as  in  the  other;  then,  adding  or 
subtracting  the  two  equations,  according  as  these  equal 
quantities  have  different  or  same  signs,  these  terms  will 
destroy  each  other,  and  the  elimination  w^ill  be  effected. 


Ex.1. 
Here  mult. 

.  (ii)  by  4, 

but 

4aj  +  7/  =  34 )     (i) 
Ay  +  X  =  1^)     (ii) 
lC>y^Ax=U, 
2/ +  42/ =  34;     (i) 

,*, 

subtracting, 
and  (ii) 

152/          =  30,  and  . 
aj=16-42/=16-8  = 

8. 

Ex.  2. 

Here 

and,  mult,  (i)  by  4, 

4x-   2/=    7)     (i) 
3ic  +  42/  =  29f     (ii) 
3aj  +  42/  =  29, 
IQx  -42/ =  28; 

.-.  adding, 

19a;           =  57,     and  .*. 

^  =  3; 

and  (i)  y 

=  Ax-  7  =  12-7  =  5. 

Sometimes  we  cannot  make  the  coefKcients  equal 
by  multiplying  only  one  of  the  equations ;  but  shall 
have  to  multiply  both  by  some  numbers,  which  it 
will  be  easy  to  perceiv^e  in  any  case. 

Ex.  3. 


Mult,  (i)  by  3, 
-  (ii)  by  2, 

2x+2>y=     A) 
Zx-2y  =  -l\ 
6a;  +  92/  =     12 
Qx  _  42/  =  -  14 

CO 
(ii) 

subtracting, 
and  (i)     2x  = 
4* 

132/=     26, 
=  4-3y  =  4-6  = 

and  /.  2/  =  2; 
-2;    .•.x  =  - 

82 


6BIPLE   EQUATIONS. 


97.  Second  riuthod, — Express  one  of  the  unknown 
quantities  in  terms  of  the  other  by  means  of  one  of  the 
equations,  and  put  this  value  for  it  in  the  other  equation. 
Ex.  4.     1x  +  \  (2y +4)  =  IG  ^     or  reducing,  35a;  +  2?/  =  76  ^    (i) 
Zy-lix  +2)=    8^  127/-    a;  =  34^   (ii) 

Here  from  (ii)  x=l2y-  34,  and  from  (i)  35  (12?/  -34)+  2y  =  76, 
whence  y  =  3,  and  .*.  ic  =  2. 
9      Third  metliod, — Express  the  same  quantity  in 
terms  of  the  other  in  both  equations,  and  put  these 
vahies  equal. 
Ex.  5.     hx-\  (5?/  +  2)  =  32  j)     or  reducing,  20a;-5?/=130 )   (i) 
Zy  +  -5  (.c  +  2)  =    9  ^  97/+  x=  25  \  (ii) 

Here  in  (i),  y  =  J  (20a;  -  130),     in  (ii)  y  =  l  (25  -  x) ; 

.-.  ]  (20a;  -  130)  =  -J-  (25  -  x),   whence  x  =  7,  y  =  2. 
The  first  of  these  methods  is  generally  to  be  preferred ;  but  the 
second  may  be  used  with  advantage,  whenever  either  x  or  y  has 
a  coefficient  unity  in  one  of  the  equations. 
Ex.  45. 


1. 

2x  +  9y  =  n) 
4x  +    y  =    5  ) 

2. 

X  +    y  ^  a 

? 

3.  2.r  -  7/  =  8 ) 
27/  +  a;  =  9 ) 

ax  +  hy  -  Ir  ^ 

4. 

ax  +   y  =  h) 
X  +  hy  =  a) 

5. 

2x-   97/ =  11) 
3a;  -  127/  =  15  \ 

6.  Ix  +  ay  =  h  ) 
ax  -ly  =  a) 

7. 

2a;  +  3?/  -  8  =  0 ; 
7x-   2/- 5  =  0^ 

8. 

ax  =  ly) 
X  +  y  =  c    ) 

9.  5a;  +  47/  =  58  J 
3a;  +  7y  =  67  i 

10. 

x(y  +  7)  =  y(x  +  l) 
2a;  +  20  =  3?/  +  l 

?    ^1- 

i 

ix^ly^Ul 
lx^{y=    5^ 

12.  ia;  +  ]7/  =  43) 
{x^iy=42\ 

13. 

X     y 
a      h 
X     y 
c      d 

14. 

.-...■^=.1 

b      c 

«±  .  ^  =  0 
c        a 

16. 

ax  +  ly  =  c*" 

17. 

18.^4-1-? 

=0i 

h  +  y      a  +  X        J 

-_^=  1 

h     a 

a     0            c 

19. 

i  {2x  +  3y)  +  Ix  = 

.?! 

20.  J(2a;- 

-y)  + 1  =  1(7 +  x) 

^(7y-3a«)-    y  = 

i 

(3- 

4j)  + 3  =  ^(5^^-7) 

SIMPLE  EQUATIONS. 


83 


21.      a;-|(y-2)  =  5^  22.  .yyiy  +  |  (a;-Cy  +  1)  =  J(x-3)| 

J(x-5y  +  8)  =  |(3»-13y)  +  ef 
23.  j\(3x  +  4y  +  3)- Jj(3x-y )  =  5  +  Ky-8| 


42/-^(«  +  10)  =  3i 


24.  2«  - 


4y  - 


-'-  (9y  +  5a;-8)-:[(a!  +  jr)  =  JyC^a;  +  C) 
2/  +  3    „    Si/  —  2x 


4^5 


8-05 


241- 


2y+l 


25.  ic- 


2/  + 


2/- 


05-18 


'=zoJ± 


3y 


99.  Simultaneous  equations  of  three  unknown  quan- 
tities are  solved  by  eliminating  one  of  tliem  by  means 
of  any  pair  of  the  equations,  and  then  the  sa7ne  one  by 
means  of  another  pair :  we  shall  thus  have  two  equa- 
tions involving  the  same  two  unknown  quantities, 
which  may  now  be  solved  by  the  preceding  rules. 

Similarly  for  those  of  more  than  three  unknowns. 


Ex.  1.  05  -  2y  +  3^  =  21 

2.r  -  Sy  +    z  =  l\ 
3o5-    y  +  2z  =  9] 
From  (i)   2o5  -  4y  +  62  =  4 
(ii)  2o5  -  3y  +    z  =  l 
-   y  +  52=3 

hence  (a)  y  =  5^  -  3  =  2, 


(ii)  Again  (i)  3.?5  -  6y  +    9^=6 
(iii)  (iii)  3o5  —    y  +    2z=0 


Ex.2. 


X     y~  r 
1 


a  c 

-  +  - 

05  Z 

I  c 

-  +  - 
y  z 


and  05  =  - 


z      p 
2pqra 


(i) 
(ii) 
(iii) 


-52/+    72=-3(/3) 
but        -  5?/  +  252=  15  (a) 
(a)      .-.  - 182  =  -T8,  and  2  =  1 : 

and  (i)  05  =  2  +  2y  -  32  =  3 : 

5_  1_1 
P 


From  (ii)  and  (iii)  -  - 

^    '  X     y      q 

;,  ...  a      h      1 

and  (1)  -  +  -  =  - 

X     y      r 

2a      1      1      1  _  ((/  +  r)jo- 

X       q      r     p  pqr 


or 


soy: 


2pqrh 


Ipqrc 


(q  +  7')p-q7*'  ^^  ^      {p  +  r)  q-pr^  "       ip  +  q)  r-pq^ 

which  latter  values  may  be  written  dovrn  at  once  from  the  Sym- 
metry of  the  equations,  since  it  is  obvious  that  the  values  of  y  and 
z  will  be  of  the  s:ame/<?r??i  as  that  of  o*,  only  interchanging  (for  y) 
a  with  6,  and  7;  with  q,  and  (for  2)  a  with  c,  and  p  with  r. 


84  SIMPLE   EQUATIONS. 

Ex.  46. 

1.  2aj  +  3?/  +  42  =  20  ^     2.  5x  +  Sy  =  C5  ^i  3.  3a;  +  2^/  -   2=20  \ 
3aj+42/  +  52  =  26            2?/-    2  =  11  2aj  +  3^/ +  6^=70 

Zx  +  52/  +  Cs  =  31  J  3ic  +  42  =  57  J  ic-    y  +  62=41  J 

4.   a;+2/  +  2=5  1      5.    x+2y=7       "|       G.  a=2/  +  2l    7.  xy=x+y     \ 
x+y=z-7  }  y  +  2z=2       }  b=x+z  \        a'2=2(a;+2)  > 

x-i=y  +  z  J  ?>x  +  2y=z-l  J  c=x+y  J        2^2=3(2/  +  2)J 

8.2(x-y)^Zz-2  ]  9.  \x^yy=\2-\z  1  10.  2/  +  l2=Ja;  +  5      1 

0^+1=3(^+2)  ^       \y^lz=  S-^ix        i(-^-lH(2/-2)=A(2  +  3) 
2^--h3.=4(l-2/)  J       1^+|2=10        J  a.'-J(27/-5)=lJ-^V  J 

Ex.  47. 
].  What  fraction  is  that,  to  the  numerator  of  which  if  7  be 
added,  its  value  is  f ;  but  if  7  be  taken  from  the  denominator  its 
value  is  J  ? 

3.  A  bill  of  25  guineas  was  paid  with  crowns  and  half  guineas  ; 
and  twice  the  number  of  half  guineas  exceeded  three  times  that 
of  the  crowns  by  17  :  how  many  were  there  of  each  ? 

3.  A  and  B  received  £5  17s  for  their  wages,  A  having  been  em- 
ployed 15.  and  B  14  days  ;  and  A  received  for  working  four  days 
II5  more  than  B  did  for  three  days :  w^hat  were  their  dail}^  wages  ? 

4.  A  farmer  parting  with  his  stock  sells  to  one  person  9  horses 
and  7  cows  for  £300 ;  and  to  another,  at  the  same  prices,  G  horses 
and  13  cows  for  the  same  sura  :  what  was  the  price  of  each? 

5.  A  draper  bought  two  pieces  of  cloth  for  £12  13^,  one  being 
Ss  and  the  other  9^  per  yard.  He  sold  them  each  at  an  advanced 
price  of  2s  per  yard,  and  gained  by  the  whole  £3.  "What  were 
the  lengths  of  the  pieces  ? 

6.  There  is  a  number  of  two  digits,  which,  when  divided  by 
their  sum,  gives  the  quotient  4  ;  but  if  the  digits  be  inverted,  and 
the  number  thus  formed  be  increased  by  12,  and  then  divided  by 
their  sum,  the  quotient  is  8.     Find  the  number. 

7.  A  rectangular  bowling-green  having  been  measured,  it  was 
observed  that,  if  it  were  5  feet  broader  and  4  feet  longer,  it  would 
contain  IIG  feet  more  ;  but,  if  it  were  4  feet  broader  and  5  feet 
longer,  it  would  contain  113  feet  more.     Find  its  present  area. 

8.  Find  three  numbers  A,  B,  C,  such  that  .4  with  half  of  i?, 
B  with  a  third  of  G,  and  C  with  a  fourth  of  J,  may  each  be  1000. 


8IMPT,E  EQUATIONS.  85 

0.  A  train  leij  Cambridge  for  London  with  a  certain  number  of 
passengers,  40  more  second-class  than  first-class ;  and  7  of  the 
former  would  pay  together  2s  less  than  4  of  the  latter.  The  fare 
of  the  whole  was  £55.  But  they  took  up,  half-way,  35  more 
second-class  and  5  first-class  passengers,  and  the  whole  fare  now 
received  was  J  as  much  again  as  before.  What  was  the  first-class 
fare,  and  the  whole  number  of  passengers  at  first  ? 

10.  A  person  rows  from  Cambridge  to  Ely,  a  distance  of  20  miles, 
and  back  again,  in  10  hours,  the  stream  iiowing  uniformly  in  the 
same  direction  all  the  time  ;  and  he  finds  that  he  can  row  2  miles 
against  the  stream  in  the  same  time  that  he  rows  3  miles  with  it. 
Find  the  time  of  his  going  and  returning. 

11.  The  sum  of  the  two  digits  of  a  certain  number  is  six  limes 
their  difference,  and  the  number  itself  exceeds  six  times  their  sum 
by  3  :  find  it. 

12.  A  grocer  bought  tea  at  10s  per  lb,  and  coffee  at  2s  6J  per  lb, 
to  the  amount  altogether  of  £31  5s:  he  sold  the  tea  at  8s,  and  the 
coffee  at  4s  (jd^  and  gained  £5  by  the  bargain :  how  many  lbs  of 
each  did  he  buy  ? 

13.  A  and  B  can  do  a  piece  of  work  together  in  12  days,  which 
B  working  for  15  days  and  Cfor  30  would  together  complete  ;  in 
10  days  they  would  finish  it,  working  all  three  together  ;  in  what 
time  could  they  separately  do  it  ? 

14.  A  sum  of  £12  18s  might  be  distributed  to  the  poor  of  a 
parish  by  giving  |  a  crown  to  each  man  and  Is  to  each  woman  and 
each  child,  or  ^  a  crown  to  each  woman  and  Is  to  each  man  and 
each  child,  or  ^  a  crown  to  each  child  and  Is  to  each  man  and  each 
woman :  how  many  were  there  in  all  ? 

15.  Divide  the  numbers  80  and  90  each  into  two  parts,  so  that 
the  sum  of  one  out  of  each  pair  may  be  100,  and  the  diflerence  of 
the  others  30. 

16.  Some  smugglers  found  a  cave,  which  would  just  exactly 
hold  the  cargo  of  their  boat,  viz.  13  bales  of  silk  and  33  casks  of 
rum.  While  unloading,  a  revenue  cutter  came  in  sight,  and  they 
were  obliged  to  sail  away,  having  landed  only  9  casks  and  5  bales, 
and  filled  one-third  of  the  cave.  IIow  many  bales  separately,  or 
how  many  casks,  would  it  hold  ? 

17.  A  person  spends  2s  Gd  in  apples  and  pears,  buying  the 
apples  at  four,  and  the  pears  at  five  a  penny ;  and  afterwards 


86  SIMPLE   EQUATIONS. 

accommodates  a  neighbour  with  half  his  apples  and  a  third  of  his 
pears  for  IM.     How  many  of  each  did  he  buy  ? 

18.  A  party  was  composed  of  a  certain  number  of  men  and 
women,  and,  when  four  of  the  women  were  gone,  it  was  observed 
that  there  were  left  just  half  as  many  men  again  as  women  :  they 
came  back,  however,  with  their  husbands,  and  now  there  were 
only  a  third  as  many  men  again  as  women.  What  were  the 
original  numbers  of  each  ? 

19.  A  and  B  play  at  bowls,  and  A  bets  B  Zs  to  28  on  every 
game :  after  a  certain  number  of  games,  it  appears  that  A  has 
won  3s ;  but  had  he  ventured  to  bet  5s  to  2s,  and  lost  one  game 
more  out  of  the  same  number,  he  would  have  lost  30^.  How 
many  games  did  they  play  ? 

20.  A  person,  being  asked  how  many  oranges  he  had  bought, 
said  ^  These  cost  me  Is  6d  a  dozen ;  but  if  I  had  got  the  five  into 
the  bargain  which  I  asked  for,  they  would  have  cost  me  2^d 
a  dozen  less.'     How  many  had  he  ? 

21.  Having  45s  to  give  away  among  a  certain  number  of  per- 
sons, I  find  that  if  I  give  3s  to  each  man  and  Is  to  each  woman, 
I  shall  have  Is  too  little,  but  that,  by  giving  2s  6d  to  each  man  and 
Is  Od  to  each  woman,  I  may  distribute  the  sum  exactly.  How 
many  were  there  of  men  and  women  ? 

22.  Find  a  number  of  three  digits,  the  last  two  alike,  such  that 
the  number  formed  by  the  digits  inverted  may  exceed  twice  the 
original  number  by  42,  and  also  the  number  formed  hy  putting 
the  single  figure  in  the  midst  by  27. 

23.  A  party  at  a  tavern,  having  to  pay  their  reckoning,  and 
being  a  third  as  many  men  again  as  women,  agree  that  each  man 
shall  pay  half  as  much  again  as  each  woman ;  but,  a  man  and  his 
wife  having  gone  off  without  paying  their  share,  lOfZ,  the  rest  had 
each  to  pay  2d  more.     AVhat  was  the  reckoning  ? 

24.  A,  B,  0.  sit  down  to  play :  in  the  first  game,  A  loses  to 
each  of  ^and  6'as  much  as  each  of  them  has,  in  the  second 
B  loses  similarly  to  each  of  A  and  C,  and  in  the  third  C  loses 
similarly  to  each  of^  and  B;  and  now  they  have  each  245.  What 
had  they  each  at  first  ? 


CHAPTEE   VIII. 

INDICES,    AND   SURDS. 

100.  It  was  stated  in  (45),  that,  wlien  any  root  of  a 
quantity  cannot  be  exactly  obtained,  it  is  expressed  by 
the  use  of  the  sign  of  Evolution,  as  V3,  V2ac,  V  d:  +  c^^ 
and  called  an  Irrational  or  Surd  quantity. 

It  was  also  stated  in  (46)  that  there  cannot  be  any 
even  root  of  ^negative  quantity ;  but  that  such  roots  may 
be  expressed  in  the  form  of  surds,  as  V-3,  V-<^^ 
^-{a'^+h'')^  and  are  then  called  im-possiUe  or  wiagi- 
nary  quantities. 

These  we  shall  considermoreatlength  in  thischapter. 

It  was  seen  in  (20),  that  powers  of  the  same  quantity  were 
multiplied  by  adding  their  indices  ;  we  shall  now  prove  this  rule^ 
to  be  generally  true,  which  was  there  only  shewn  to  be  true  in 
particular  instances. 

101.  To  prove  that  a^^  x  a^=ra"^+*^,  iche7i  m  a7id  n  are 
any  positive  integers. 

Since  by  (9)  a''^^  =  a  x  a  x  &c.  {m  factors) 
and  a^  =  a  X  a  X  &c.  (71  factors), 
it  follows  that 

a'^xa'^=^axax &c.  (m factors)  xaxax &c. {n factors) 
=:a xax &c.  {ni+n factors)  =:a*^+%  by  (9). 

102.  Hence  (c^'^)^^=a^'^=(^^^)"^; 
for(<^'^)^=a'^.^'^.a^.&c.  n  factors=a'"+^^+*^+*^°-  ^^^^'^'z^a""*, 
and(a^)^=a^a".a^&c.  m  factors=a"+"+«+^°-'"^^^°^^=a^''* ;  - 

.-.  since  a^^^r^"""',  we  have  {c(/^Y^a''^^={a'y^  \ 
that  is,  the  n^^  power  of  the  m}^  power  of  a  =  the  m*^ 
power  of  the  n^^^  power  of  a,  and  either  of  them  is  found 
by  multiplying  the  two  indices. 

103.  Hence  also  Vc^^^CV^')"": 

for  let  V  a'''=x''\  then  a^=  {x"')''=:  (i^«)"*  by  (102) ; 
hence  a  —  a?^,  and  .*.  \/a  =  x,  and  (V  ci)"^  =  ^''^; 


88  INDICES, 

but  also,  by  our  first  assumption,  \l a^  =  a;*"; 

hence  we  have  yoJ""  =  (^  ciY' ; 
that  is,  the  ii^^'  root  of  the  m^^  power  of  2^.^  the  m^^ 
power  of  the  n^^  root  of  a. 

lOi.  These  results  refer  as  yet  only  to  positive  in- 
tegral indices,  which  (9)  were  first  used  to  express 
briefly  the  repetition  of  the  same  factor  in  any  product. 

But  now,  suppose  w^e  WTite  down  a  quantity,  with  a 

jp 
i^'O^iiiWQ  fraction  for  an  index,  such  as  a^,  and  agree  that 
such  a  symbol  shall  be  treated  by  the  same  law  of 
Multiplication  as  if  the  index  were  an  integer^  viz. 
cC\cf'  =  t^^'+": — what  would  such  a  symbol,  so  treat- 
ed, denote? 

Since  it  follows  from  this  law,  in  the  case  of  positive 
integers^  that  ici'^^y^  =  a"^"',  we  should  have  here  also 

{a^yz=:i'i=a'P\  and  hence  it  appears,  that  cif^  would 
denote  such  a  quantity  as,  when  raised  to  the  c^^power^ 
becomes  equal  to  a^.  But  that  quantity,  whose  q**^ 
power=:a^,  is  (10)  the  q^^  root  of  oP ;  and,  therefore, 

a^  z=z  ya'P,  or  =  (V  ay  by  (103). 

Hence,  when  a  fractional  index  is  employed  with 
any  quantity,  the  nicmerator  denotes  ^  power ^  and  the 
denominator  a  root  to  be  taken  of  it. 

Thus  a^  =  2°"*  root  of  1*'  power  of  a  =  V  «j  «*  =  V<^j  ^^=V  «)  ^^ 

€?  -  cube  root  of  square  of  a  =  V^'^^ 
or      =  square  of  cuhe  root  of  a  =  (^  ay  ; 
so  a^=\la'^  or  (V«)^   a^=a^=a^=&c.,  or   ^a=*/a'=ya''=&c. 

105.  Again,  if  we  write  down  a  quantity  with  »? 
negative  index,  as  a~^  (where  p  may  now  be  integral 
or  fractional),  and  agree  that  this  symbol  shall  be 
treated  by  the  same  law  of  Mult"  as  if  the  index  were 
positive,  what  would  such  a  symbol,  so  treated,  denote? 


AND   SURDS.  89 

Ey  this  law  we  should  have  a^'-^^  x  a~P=a"»+p-^=a^ ; 

but  ^VQ  have  also         a''^-^^'  -?-  a^^  = =  — -—  =a"' : 

so  that,  to  multijjly  by  a"^,  is  the  same  as  to  divide  by  ccP: 

and,  therefore,  1  x  cr-^  =  1  -^  a^,  or  <z"-p  =  — . 

Hence,  any  quantity  with  a  negative  index  denotes 
the  recijyrocal  of  the  same  with  the  samo^J^ositive  index. 

Thusa-=1,    «-=!,   «-*=!=  i,  or  =  V«-'=^/J; 


7,-i  = 


11  •>,-.    VI 


Hence  also  any  power  in  the  numerator  of  a  quan- 
tity may  be  removed  into  the  denominator,  and  vice 
versd^  by  merely  changing  the  sign  of  its  index. 

Thus  «-iv-  =  ?:5: = -fC = 5!£: = &c. 

c         o~'c         a** 

106.  Lastly,  if  we  write  down  a  quantity  with  zero 
for  an  index,  as  a\  and  agree  that  this  symbol  shall 
be  treated  as  if  the  index  were  an  actual  number, — 
what  then  would  it  denote  ? 

Since,  by  this  law,  cd"  x  <^^=a°+'''=^^,  it  follows  that 
a°  is  only  equivalent  to  1,  whatever  be  the  value  oi  a. 

In  actual  practice,  such  a  quantity  as  a° "would  only  occur  in  cer- 
tain cases,  where  we  wish  to  keep  in  mind  from  what  a  certain  num- 
ber may  have  arisen:  thus  (a^  +  2fl^^  +  3a+&c.)-f-a^=a+2+3a"'^  +  &c., 
vrhere  the  2  has  lost  all  sign  of  its  having  been  originall}^  a  coeff.  of 
some  power  of  a  \  if,  however,  we  write  the  quotient  a+2a°  +  3a~^+ 
<S:c.,  we  preserve  an  indication  of  this,  and  have,  as  it  were,  a  con- 
necting link  between  the  positive  and  negative  powers  of  a. 
P 

The  quantity  a^  is  still  called  a  to  the  power  of  -,  and  similarly 

in  the  case  of  «~^,  a° ;  but  the  v^ox^  power  has  here  lost  its  original 
meaning,  and  denotes  merely  a  quantity  icitJi  an  index,  whatever 
that  index  may  be,  subject,  in  all  cases,  to  the  Law,  a*^^a«=a»^+'^. 


90  INDICES, 

Ex.  48. 

Express,  Y:ii\\  fractional  indices, 

2.  a  VJ«  +  (V«)'  +  Va«6  +  V^' ;  VaW  +  a  CV^)*'  +  Va^°+  V»^. 
Express,  with  negative  indices,  so  as  to  remove  all  powers, 
(i)  into  the  numerators,  and  (ii)  into  the  denominators, 

^1      2       3      4«     55     a»      3a'     5a     4&     25' 
a      5-       c^        5        a       5^        5         h'      a^        a^ 

a^        ^      ^      J^.    _^      ?^  ,         ^  5g 

3Z;V  ""  ^  "^  T  "^  3a5c'    2V^  "^  3V^'  "  4Va'^  "^  a  V^'* 
Express,  with  the  sign  of  Ecolution^ 

Express,  vfiih  positive  indices,  and  with  the  sign  of  Evolution, 

-2         -1-'*         _3S_5 

6.  a~^5c+a5~'c+a~^5~^c~^  +  a~''6~'V  ;  a  ^+atb  "^'+a  -b^+h  ^' 
a-^-^        2a        Zh-'c-''  1  ^'      L!      ^     £^ 

107.  It  follows,  then,  that,  whatever  he  the  indices, 
;  awi  1 

SO  that  (i)  to  multiply  any  powers  of  the  same  quan- 
tity, we  must  add'  the  indices,  (ii)  to  divide  any  one 
power  of  a  quantity  by  another,  we  must  subtract  the 
index  of  the  divisor  from  that  of  the  dividend,  and 
(iii)  to  obtain  ViXij 2>ower  of  ajpower  of  a  quantity,  we 
must  multiply  together  the  two  indices. 

1  3^x  11-3         -1.+  3         _l 

Thus  a'xa^=a'~'- fl.  a*-^a -=  a     ^^- a"^^  a^-^  a'^=a^   ^=ai^ 
Hab-Wab  j '  =   I  ah' .  ah^  V  =  (a^5'^")  '  =  a'b*. 


AND  SURDS.  91 

Ex.  1.  Multiplication, 

a^  +  a^l^  +  o^lfi  +  ah  +  a^b'^  +  h^ 


A   1.  S.  1  4  15 

-  ahi  -  a'b^  -  ah  ~  al^  -  a^b^  ~  ^' 


a'    * 

*                         *                   *                      5^           _   52 

Ex.  2.  Division. 

a;2  -  4f<a;'^  +  26^2)  x^ 

-  a^o;"  -  4aaj'2  +  Ga%  -  2ft'2-^  (a;  - 

-A* 

5 
aj2 

n            3 

-  4ax'2  +  2a^x 

-  a^o;''  +  Aa^x  -  9.a''x^ 

-  (^x"  +  Aa^x  -  'la'^x^ 

It  is  well  to  observe  that  no  algebraic  operation  with  homogene- 
ous quantities  can  destroy  the  homogenity  (59),  which  will  be 
found  existing  throughout  in  all  the  products,  remainders,  quo- 
tients, &c.  Moreover,  in  all  such  products  and  quotients,  the  Law 
of  Dimensions  will  be  observed,  as  indicated  by  the  formulae, 
a*"  X  a"  =  «"*+",  a*^ -i- a"  =  61'"''* :  thus,  in  Ex.  2,  the  quotient  is 
of  4-|=l  dimension,  and  all  the  products  of  ^  +  1'=  I  dimensions. 
This  observation  will  often  help  us  to  detect  errors  in  Mult",  Div'', 
&c.,  especially  in  dealing  with  fractional  indices. 
Ex.  49. 

1.  Simplify  \{a-W-f'\'^\  vVTV^  ^^r/ Va^"V^  \^  a''bM'cfHr^\\ 

2.  Simplify  {x'hj.  (xy'^y^.  {^-'yV^W     {^V*  V(^*2/^  V2/*)P. 

3.  Simplify  ^{xy^ ^ xyz ^Vi^V' [ '^,  Vo^^^^'+V^ x  ^a^J-^V'*-'^, 

4.  Multiply  a;  +  2y-  +  3s"  hy  x  -2y'^  +  3^^ 

5.  Multiply  a^  +  aH^  +  c^h  +  b'^  by  a^  -  h^, 

6.  Multiply  a^  -  2a''b^  +  4ah^  -  Sab  +  Uah^  -  326^  by  a- +25^ 

7.  Multiply  a'-^  -a^  +  a'^  —  «  "2  by  (^2  +  a^  -  c^"  —  a  2, 

8.  Multiply  ic^  +  x'^i/'^  +  a?^?/'*^  +  aj^y'"^  +  x^y'^  +  y'^ 

by  a;**  —  x^y  *  +  aj''y     —  y    . 


92  INDICES, 

9.  Dividcl6aj-y-by2a5^-2/  ,  and  x'^-y^  by  x"^  -  y'^, 

10.  Divide  a'^  -  ^iV  by  a"^  +  2h^,  and  x  -  2x^  +  1  by  a;^-2.5^+l. 

11.  Divide  Sa^  +  V^^  -  c  +  GaV^c^  by  2a^  +  J"^  -  A 

12.  Find  the  cubes  o£  a'-^b  ^  +  a''^Z»  and  Ix^y'"  —^x'^y^, 

13.  Find  the  cube  of  J  -  2a* 5«  +  Zh^\ 

si  -12 

14.  Write  down  the  square  of  a''^  -  2a^  +  3  -  2a  ^  +  a'^, 

15.  Find  the  fourth  and  fifth  powers  ofx^-y^,  and  ah'^-ah^. 

16.  Find  the  square  root  of  a'^b  ^  +  2aZ>'^  +  3  +  2a~^b  +  a'^b'^, 

17.  Find  the  square  root  of 

at-  _  3a  +  3^-a^  -  21a^  +  45  -  C3a"^  +  90a'^  -  108a-^  +  81a"i 

18.  Find  the  cube  root  of 

a  ^a;2  -  Zar'^x  +  Ga  -o;^  -  7  +  ^a^x  ^  -  3aaj"^  +  a-x  ^. 

19.  Find  the  fourth  root  of  x^y'^-ix^y"^  ■\-^xy^-Ax'^y's  -^xY" 

9     3  3  3     9 

20.  Find  the  fourth  root  of  lGaj"-96a;22/T+21Ga;V2-21Ga:2y* +8I2/'. 


108.  SincG  every  fractional  index  indicates  by  its 
denominator  a  root  to  be  extracted,  all  quantities  hav- 
ing sucli  indices  are  expressed  as  surds. 

When  a  oiegative  quantity  has  the  denominator  of 
its  index  (reduced  to  its  lowest  terms)  even  (46),  the 
expression  will  be  imaginary. 

Thus  ^-3  or  (-3)-^,  ^-9  or  (-9)^,  are  imaginary  quantities; 

but  (  -4)^  is  not  so,  since  it  is  the  same  as  (-4)^,  where  the  root 
to  be  taken  is  odd. 

109.  In  the  case  of  tx  numerical  surd,  expressed  with 
a  fractional  index,  should  the  numerator  be  any  other 
than  imiti/y  we  may  take  at  once  the  required  power, 
and  so  have  unity  only  for  the  numerator,  and  a 
simple  root  to  be  extracted. 

Thus  2^  =  (2^)'^  =  4^  or  V4,  3"?  =  (3-')^  =  (^V)*  or  VoV- 


AND   SURDS.  93 

110.  Quantities  are  often  expressed  in  the  form  of 
surds,  wliicli  are  not  really  go,  i,  e,  when  we  ccm^  if 
we  please,  extract  the  roots  indicated. 

Thus  ^Ja^  V7.  (a'+a5  +  &^)*  are  actually  surds,  whose  roots  we 
cannot  obtain ;  but  -.y<^^,  y27,  (4fi^  +  Aal)  +  P)-  are  only  appa- 
rently  so,  and  are  respectively  equivalent  to  «,  3,  2a  +  h. 

Conversely,  any  rational  quaiitity  may  be  expressed 
in  the  form  of  a  surd,  by  raising  it  to  the  power  indi- 
cated by  the  denominator  of  the  surd-index. 

Thus  2=4^=VS='S:c.,  a=\la\  fc=  (|c^)^,  a+x=  (a?  +  2ax  +  X')^ 

111.  In  like  manner  a  mixed  surd,  ^^  e.  a  product 
partly  rational  and  partly  surd,  may  be  expressed  as 
an  entire  surd,  by  raising  the  rational  factor  to  the 
power  indicated  by  the  denominator  of  the  surd-index, 
and  placing  beneath  the  sign  of  Evolution  the  prod- 
uct of  this  power  and  the  surd-factor. 

Thus  2V3  =  V4  X  v3  =  V12:  3.2^  =  3  yi  =  V^^  x  V^  =  V108, 

2a^h  ^  V4«^6,  Aa  |/~^  =  y''-^  =  V32c.^c. 

Conversely,  a  surd  may  often  be  reduced  to  a  mixed 
form,  by  separating  the  quantity  beneath  the  sign  of 
Evolution  into  factors,  of  one  of  which  the  root  re- 
quired may  be  obtained,  and  set  outside  the  sign. 

Thus  V20  =^/I1^5  =  2^5,  y24  =  VsVs  =  2r/3, 

Vfl^  =  ^a-^^,  V|K^  =  tadV2a^, 

112.  A  surd  is  reduced  to  its  simplest  form,  when 
the  quantity  beneath  the  root,  or  surd-factor,  is  made 
as  small  as  possible,  but  so  as*  still  to  remain  integral. 

Hence,  if  the  surd-factor  be  Vi  fraction,  its  nwmJ  and 
den""  should  both  be  multiplied  by  such  a  number,  as 
will  allow  us  to  take  the  latter  from  under  the  root. 
/2       72.3      1    ^^  5    s  /24     ^3/3-3  /3.5^    ,  __ 


94  INDICES, 

These  latter  forms  allow  of  our  calculating  more  easily  tho 
numerical  values  of  the  surd  quantities.  Thus  to  find  that  of  ^f, 
we  should  have  had  to  extract  both  ^2  and  y'o,  and  then  to  divide 
the  one  by  the  other,  a  tedious  process,  since  each  would  be  ex- 
pressed by  decimals  that  do  not  terminate  ;  whereas  in  ^^6,  we 
have  only  to  find  ^6,  and  divide  this  by  the  integer,  3. 

Similar  surds  are  tliose  which  have,  or  may  be 
made  to  have,  the  same  surd-factors. 

Thus,  Z^a  and  *Ja^  2a  \/c  and  3&  y^,  are  pairs  of  similar  surds ; 

and  ^8,  -^50,  ^18  are  also  similar,  because  they  may  be  written 

2V2,  5V2,  3V2. 

Ex.  50. 

3  2  3  3  1  3 

1.  Express  4*,  9^,  3'^,  2"^,  (|)"^,  (i)"  ^  with  indices,  whose 
numerator  is  unit}^ 

2.  Express  5, 2i,  f «,  |a",  l(a  +  ?>),  as  surds,  with  indices  ^  and  J. 

3.  Express  3"^,  (3J)-^,  «-^  ab'^c-^^  with  indices  J  and  -  {, 
Reduce  to  entire  surds 

4.  5V5,  yi  f.S*   yill{\)-\  25  (l|)-i 

5.  3V2,  8.2- J,  4,2»,  S.S"?,  f  (§)-^,  i  (J)"?. 

6.  2Va,   7aV2i,  »  («^)-')  (»  +  *)  («'  -  ^°)'  ^  («  -  ^)  ('^'  -  ^')-'- 

„  /25    „  /2«      2a'    /3J     2n!3   /  9      ,       ,      /«-« 

7.  a  y^-,  3ax  ^/g-,    -^-^^^ -,    -^^,  (a.x)  ^/— . 

Reduce  to  their  simplest  form 

8.  V45,  V125,  3V432,  V135„3V432,  Vi,  2^%  3Vi,  4^3?. 

9.  8S  32'^,  72^,  (li)-2,  (201)-V(30S)-UVV,5V-^.,|^9J. 

10.  Shew  that  V12,  3v75,  ^Vl47,  f  VA,  Vrc.  and  (144)"^^  are 
similar  surds. 


113.  To  co.mpare  surds  with  one  another  in  magni- 
tude, express  them  as  entire  surds,  and  then  reduce 
their  indices,  if  necessary,  to  a  common  denominator, 
simplifying  as  in  (109) :  their  relative  vahies  will  be 
Ti0%v  apparent. 


AND   SURDS.  95 

Thus  3  >^2  and  2  -^3,  expressed  as  entire  surds,  are  -^18  and  Vl2, 
and  it  is  at  once  plain  which  is  greatest :  but  3  ^1  and  2  ^3^  or 
their  equivalents  >^18  and  ^24,  in  which  different  roots  are  to  be 
taken,  cannot  be  at  once  compared;  here  then  182=18^=  ^5832, 

1  2 

and  24^=24^=^576,  and  now  their  comparative  values  are  evident. 

114.  To  add  or  subtract  surds,  reduce  them,  when 
similar,  to  the  same  surd-factor,  and  add  or  subtract 
their  rational  factors. 

Thus    V^  +  V^O  -  Vl8  =  2  V2  +  5  V2  -  3  V2  =  4  V2, 
4aV'^^&V8^-\/i25^*=4a=5  V^+2a^5  ^&-5a^6 \/h=a'h\/h. 

Dissimilar  surds  can  only  be  connected  by  their 
signs. 

115.  To  multiply  surds,  reduce  them  (113)  to  the 
same  surd-index,  and  multiply  separately  the  rational 
and  surd  factors,  retaining  the  same  surd-index  for 
the  jproduct  of  the  latter. 

_Thus  V8  X  3  V2=3  VlG=12,  2  V3  x  3  VlO  x  4  ^0=24  Vl80=144  V5 
2V3  X  3V2=2V27x3V4  =  GV108. 

Compound  surd  quantities  are  multiplied  according 
to  the  method  of  rational  quantities. 

Ex.  1.  (2  ±  V3)'  =  4  ±  4^3  +  3  =  7  ±  4  V3. 
Ex.  2.  (2  +  V^)  (2  -  V3)  =  4-  3  =  1. 

Ex.  3.  (2  +  V3)  (3  -  V2)  =  6  +  3  V3  -  2  V2  -  V^. 

Ex.  4.  (1  +  V2)*  =  1  +  4  V2  +  12  +  8  V2  +  4  =  17  +  12  V2. 

116.  Division  of  surds  is  performed,  when  the  divi- 
sor is  a  simple  quantity,  by  a  process  similar  to  that 
for  multiplication. 

_Thus(8V2-12v3  +3v6-4)-5-2v6  =  4Vf-6vf +  i-  4 

=  jV3-3v2+i-W6> 

(2  V3  -  6  V2)  -*-  V^  =  2  Vf  -  eVatr  =  V2  -  V864. 


96  INDICES, 

117.  But,  if  the  divisor  be  compound^  the  division  is 
not  so  easily  performed.  The  form,  however,  in  which 
compound  surds  usually  occur,  is  that  of  a  iinomial 
quadratic  surd,  i,  e,  a  binomial,  one  or  both  of  whose 
terms  are  surds,  in  which  the  square  root  is  to  be 
taken,  such  as  3  +  2  |/o,  2  |/3-3  4/^5  or,  generally, 
Vet  :k  |/J,  w^here  one  or  both  terms  may  be  irrational ; 
and  it  will  be  easy,  in  such  a  case,  to  convert  the 
operation  of  division  into  one  of  multiplication,  by 
putting  the  dividend  and  divisor  in  the  form  of  a 
fraction,  and  multiplying  both  num"^  and  den'^  by  that 
quantity,  which  is  obtained  by  changing  the  sign  be- 
tween the  two  terms  of  the  den^  By  this  means  the 
den'  will  be  ycl^Aq  rational :  thus,  if  it  be  originally  of 
the  form  \Ul  i  4/5,  it  wall  become  a  rational  quantity, 
a-b^  when  both  num'and  den' are  multiplied  by  V<^±V&' 
2+V3  ^  (2-fV3)  (3-V3)  _  6  +  3v3^2v3-3  _  S  +  yS 
3  +  V3      (3  +  V3)  (3-V3)~  9-3        "         6    * 

P,  o   __J ^2V2-^V3^2V2^V3 

"•  2  V2  -  V3         «  -  3  5 

Fractions  thus  modified  are  considered  to  be  reduced 
to  their  simplest  form,  for  the  reason  mentioned  in  (112). 
Ex.  51. 

1.  Compare  GyS  and  4v7;  3^3  and  2  \/lO\   2  \/\^,4:\/2, 

and  3  Vo  ;  V^  and  ^11 ;  -JV^  and  ^  \/21 ;  V5,  2  Vf ,  and  3  {A\)'K 

2.  Simplify  Vl28  -  2  V^O  +  ^12  -  ylS,  ^40  -  i  V320  +  »/135. 

3.  Simplify  8  V|-i  V12  +  4  V27-2  VA,  V72-3  »/|4-6  V2H. 

4.  jNIultiply  3  V8  by  2  V6,  3  ylS  by  4  V20,  and  2  V4  by  3  V54. 

5.  Find  the  continued  product  of  3^8,  2^6,  and  3^54;  and 
cf2v24,  3V18,  and4V24. 

G.  Multiply  3  V3  +  2  V2  by  ^2>-^2,  and  2^1^-^(S  by  V^  +  V^- 

7.  Find  the  continued   product  of  4  +  2  V2,  1  -  yS,  4 -2  V2, 
V2  +  V3,  1  + V^,  andV2-V3. 

8.  Div.  2V3  +  3V2+V30  by  SyG,  and  2V3  +  3V2+V30  by  3v2. 


an£)  surds.  97 

9.  Rationalize  the  denominators  of 

1___  4         3_        8-5V2     3-f  V5  4v7-f  3v2 

2V2-V3'     V^-^'     V5  +  V2'     3-2V2'     Z-^V  b^2^2^1 

10.  Divide  2  +  4  V"   by   2  ^7  -  1,   3  +  2  V^  by  2  ^5  -  1,  and 
5  -  2  VG  by  6  -  2  V^. 

Simplify 

Va+ii'+Va-a;  1  1  ic+Vic^-1      x-^/x^-\ 

11.  -/-=^ 


-s/a+x-^a-x  a-^a^-x^      a^-vaF-x"^  x-'s/x'^-l      x-^^x^~\' 
Vi^+V.T^     VivT-V^^  ,1  1  1 

12       , -^=^   +---r.rr= ;r=:=r,  &  - 


'V.ij'^  +  l-Va;^-!      V^^'+l  +  V^^-l'     4(1  +  V^)    4(1-V.?^)   2(l+ic)- 


18.  The  following  facts  should  be  noticed. 

(i)  The  product  of  two  dissimilar  surds  caniwt  he 

rationed. 

Let  ^x  X  Vy  =  m.^  a  rational  quantity  ;  .*.  xij  ~-  m^ ; 

,  ^;^^      m'*  -  VI 

hence  ?/  =  —  =  -^^a?,  and  Vy  =  — Vti', 

or  Vy  may  be  made  to  have  the  same  surd-factor  as 
\lx ;  that  is,  ^Ix  and  \ly  must  be  similar  surds  (112). 

(ii)  A  surd  cannot  equal  the  sum  or  difference  of  a 
rational  quantity  and  a  surd^  or  of  two  dissimilar 
surds. 

For  let  ^/a  =  x  ±  Vy,  .*.  (^  =  a^'  ± 2x -^y  +  y\ 

whence  i  '^x\ly  —  a  -x^-y^  and  ±  ^Jyz= — ^^ 

or  a  surd  =  a  rational  quantity,  Avhich  is  absurd. 
,     Again,  let  \la  =  \/x  ±  Vy,  .\a  =  x  ±2  \lxy  +  y, 
whence  ifc  2  -Jxy  :=z  a-x-y^  and  ±  ^xy  =  ^  ((^-x-y)y 
or  the  product  of  two  dissimilar  surds  =  a  rational 
quantity,  which  is  impossible. 

(iii)  7/^  a  +  Vb  =  X  +  Vy,  the'?i  a  =  x,  a7id  Vb  =  Vy. 
For  since  a  +  ^/b  =x-{-^y,  we  have  V5=  {x-a)+Vy ; 
so  that,  if  ccbe  not  equal  to  a,  we  shall  have  V5=  sum 
5 


I 


98  INDICES,    AND   SUKDS. 

of  a  rational  quantity  and  a  surd, which  is. impossible*, 
hence  a?  =  <2,  and  .*.  V5  =  Vy. 

Hence  also,  \i  a  +  -Ji  =a?+V2/j  then  a-\/l=x-^y  ; 
and,  if  <^  +  ^h—0^  we  must  have  separately  (3^=0,  and 
J  =  0  ;  otherwise  we  should  have  V&  =  -  ^j  or  a  surd 
=  a  rational  quantity. 

(i v)  If  Va  4-  Vb  —  X  +  Vy,  then  Va-V  b  =  x -  V y. 
For  since  V^  -j-  V5  =  a?  +  Vy,  we  have,  squaring, 
a  +  -Jh=d'-\-2x-Jy-\-y\  ,\a—x^-\-y^  and  V5=2a?Vy; 

wdience  a-^'b  =  x'-2x  Vy + Vy,  and  ^a-s/h  =^  x-  Vy 
So  also,  if  V(^  -{-  ^5=V^+ Vy,  then  Vc^  -  Vi  =  V^-Vy 
119.  :Z(>  extract  the  square  root  of  a  hinomial  surd^ 

one  of  whose  terms  is  rational^  the  other  a  quadratic  surd. 
Let  a  +  y5  represent  the  given  surd ; 

assume  ^a  ■\-  4b  =^  \lx-\-  Vy,  .'.  '^a-^b  =^  \fx  —Jy  ; 

hence,  multiplying  these  equations,        ^d'-b=x-y  \ 

but,  since  a+^b=x  +  y  +  '^^xy^  ,\  also(118,  iii)a=cc+y ; 

.-.,  adding  and  subtracting,  a+'^a^-b^^^^x,  a-^a'^-b=^2y ' 
.\x  =  i{a  +  ^d'  -  b\  y  =  *  {a-^a'-b), 

&V(^±V^')=Va?±Vy=VB(^+^^^±VB(^-^^l- 
Ex.  Find  the  square  root  of  7  ±  2  -^10. 

Let  V7  +  2vio  =  v^  +  Vy)      .-. V7-2vio  =  va;-vy; 

and    V49  -  40  =  ic  -  y,  whence  3  =  a;  -  y ; 

but,  since  7  +  2  -^10  =  a;  +  2/  +  2  Va^y,        .•.  also  7  =  a;  +  y ; 
.-.  10  =  2a;,  4  =  2y,  or  a;  =  5,  y  =  2;  and  V7  ±  2 ^10  =  V^  ±  V^- 

Ex.  52. 

Find  the  square  roots  of 

1,  4  +  2  V3.        2.  11  +  6  V2.      3.  8  -  2  Vl5.  4.  38  -  12  VlO. 

5.  41  -  24  V2.     6.  2}  -  V^.         7.  4  J  -  J  V^-  8.  j  J  J  -  \  ^2, 

Find  the  fourth  roots  of 

9.  17  +  12  V2.  10.  5G-24V5.    11.  fV^  +  31.  12.  48^V+W15. 


CHAPTEE    IX. 

QUADRATIC     EQUATIONS. 

120.  Some  equations  involving  surds  are  reducible 
to  simple  equations,  as  in  the  following  examples. 

Ex.  1.  Vl2T^  =  2  +  V-^- 
Squaring,  we  have  12  +  x=4:+iyfx+x  .\  4V^=8,  and  V^=2j  or  ic=4. 

Ex.  2.  S  +x-  Vif^  +  y'  =  2. 

Here  Va;^  +  9  =  1  +  a* :  [observe  in  other  similar  cases  to  take 
this  step,  when  possible,  by  which  we  get  the  surd  h^  itself  on 
one  side,  and  so  it  will  disappear  upon  squaring:] 

hence  x"^  +  9  =  1  +  2x  +  x"",  and  x  =  4. 

Ex.  53. 


1. 

V5  (a;  +  2)  =  V5a;  +  2. 
1 

2. 

4. 

G. 

8. 
10. 

^/xh  +  V^  («  +  ic)  =  x'^' 

o. 

V&a;  +  a;^  =  1  +  a*. 

5. 

I'^llx  -  26  +  ^  =  l^Jj-. 

a  -\-  X  —  Va^  +  a;^  =  h. 

7. 

Va?  -  0^  =  ^aj  +  V&  +  a;. 

-  -^aj  +  V  aj  +  2V«a;  +  a^=  V^- 

9. 

a  +  X-  \l2ax  ■¥  x"^  -l. 

a  +  X  +  V»^  +  Jx  +  a;'  =  5. 

121.  QiiadraiiG  Equations  are  those  in  which  the 
square  of  the  unknown  quantity  is  found.  Of  these 
there  are  two  species : 

(i)  Pure  Quadratics,  in  wliich  the  square  only  is 
found,  w^ithout  tlie  first  power,  as  a?^  —  9  =  0,  &c.  ; 

(ii)  Adfected  Quadratics,  where  the  first  power  en- 
ters as  well  as  the  square,  as  aj""  —  3i»  +  2  =  0,  &c. 

122.  Pure  Quadratics  are  solved,  as  in  simple  equa- 
,tions,  by  collecting  the  unknown  quantities  on  one 
side,  and  the  known  quantities  on  the  other.  We 
shall  thus  find  the  value  of  cc',  and  thence  the  value 
of  cc,  to  which  we  must  prefix  the  double  sign  (±). 


100  QUADRATIC   EQUATIONS. 

Such  equations  therefore  will  have  two  equal  roots, 
with  contrary  signs. 

Ex.  1.  ic'  -  9  =  0.     Here  x"  =  9,  and  a;  =  ±  3. 

If  wc  had  put  ±x  =  ±  3,  we  should  still  have  had  only  these  two 
different  values  of  a;,  viz.  a;  =  +  3,  a;  =  -3;  since  -  jc  =  +  3  gives 
aj  =  -  3,  and  -  a;  =  -  3  gives  ic  =  +  3. 

Ex.  2.  \  {Zx"  +  5)  -  -J  {x"  +  21)  =  39  -  5x\ 

Reducing,  121a;'  =  1089 ;  .-.  a;'  =  9,  and  a;  =  ±  3. 

^       _    ^/a"^  +  x"^  +  X      h        T^         ,„-      .^  -/a^  +  x^      h  +  c 

Ex.  3. =  -  .     Here  (85.  vO =  r ; 

Va=  +  aj^-a;     ^  ^  ^~^ 

a^  +  x""      fb  +  cV       ^  x^       (7)  -  cy  a  (b  -  c) 

.*.    ^—  = ,  and  —  =  -^-r^^ —  5  or  a;  =  ±     ,,    ._  ^ . 

x^  \b  -  cj  a^         4bc      '  2^^^     ' 

The  above  method  of  reduction  from  (85.  vi)  may  alwaj^s  be  ap- 
plied with  advantage  to  an  equation  of  the  above  form,  wVien  the 
unknown  quantity  does  not  enter  in  both  sides  of  it. 

Ex.  54. 

1.  ia;'=14-3a;^     2.  aj'  +  5=-V-aj'-16.    3.  (a;  +  2)^=4a;  +  5. 

4.  , +  = =  8.  5.  J-  -  — -  =  ^     6,  8x  +  -=  -_—  . 

1  +  a;   1  -  a;       4a;''   6a;'   3  x       7 

3a;'   15.2;'  +  8  _  ,  ^    ^  a;'   a;'  - 10   _  50  +  a;' 

<.  — i ^ =  ^X     —  O.       O.  -z 7"= =  7  —  -r- , 

4     6               5     15        25 
0  ^^^     90  4-  4.7;'  _        4a;'+5  _  2a;'- 5  _  7a;'-  25 
*  a;' +  3"^  a;' +  9  ~          10     15~~   20   * 
10.z;'  +  17   12a;'  +  2  _  5a;'  -  4     14a;'+16   2a;'  +  8  2a;^ 
18     11a:' -8  ~   9   '       21     8a;'-ll~T- 
2        2  _     1 1 


13. z  + -X  14 _ 

a;+V2-i'      a;-V2-a.»      *  '  «_Va'-x'      a+^a'-x^     a;'"* 

15   V^-g-V^^"^^"_c  ^.  , Tig' 

Va'-x'  +  V6'"+;r'  "  d  *  ■^^-  ^+Va'  +  a;'=  :;/^r^/ 


123.  An  adfecied  quadratic  may  always  be  reduced 
to  the  form,  cc'+paJ+^'^O,  where  the  coeflF.  of  cc'  is  +1, 
and^,  g',  represent  numbers  or  known  quantities. 

Now,  in  this  equation,  we  have  ^  -\-jpx  =  -<7,  and, 
adding  {^jpf  to  each  side,  we  get  a;'+pa?+J^/=ij9'  -jt 


QUADRATIC  EQ.JJAJTONS,  '  TCI 

by  this  step,  tlie  first  side  becomes  a  complete  square ; 
and  taking  the  square  root  of  each  side,  prefixing,  as 
before,  the  double  sign  to  that  of  the  latter,  we  liave 


x  +  ijy=  ±  '^\f-(i^  and  x  =  -\p  ±  ^\xf-q^ ; 

which  expression  gives  us,  according  as  we  take  the 
upper  or  lower  sign,  two  roots  of  the  quadratic. 

124.  From  the  preceding  we  derive  the  following 
Hule  for  the  solution  of  an  adfected  quadratic  : 

Keduce  it  to  its  simplest  form  ;  set  the  terms  involv- 
ing iz?^  and  X  on  one  side,  (the  coeff.  oix^  being  +1?)  and 
the  known  quantity  on  the  other ;  then,  if  we  add  the 
square  of  half  the  coeff.  of  xto  each  side^  the  first  will 
become  a  complete  square  ;  and  taking  the  square  root 
of  each,  prefixing  the  double  sign  to  the  second,  we 
shall  obtain,  as  above,  the  two  roots  of  the  equation. 

Ex.  1.  x"  --Cix  =  7.     Here  ic^  -  Ga;  +  9  =  7  +  9  =  16 ; 
whence  ic  -  3  =  ±  4,  and  0^  =  3  +  4  =  7,  oraj  =  3-4  =  -l 
so  that  7  and  -  1  are  the  two  roots  of  the  equation. 

Ex.  2.  x"  +  14^  =  95.     Here  x"  +n4a;  +  49  =  95  +  49  =  144  ; 
whence  a;  +  7  =  ±  12,  and  a;  =  -  7  +  12  =  5,  or  a;  =  -  7  -  12  =  -  19. 


Ex.  55. 

1.  x'- 

-2^=8. 

2.  x''  +  lOjj  =  -  9. 

3.  x^-Ux^i20. 

4.  x-"- 

-  12^  =  -  35. 

5.  a:^  +  32a;  =  320. 

G.  cc^  +  lOOaj  =  1100. 

125.  If  the  coefiicient  of  x  be  odd^  its  half  will  be 
a  fraction.  In  adding  its  square  to  the  first  side,  we 
may  express  the  squaring,  without  efi'ecting  it,  by 
means  of  a  bracket. 

Ex.  1.  x'-^x^  -G.  Here  x''-hx^  (|)2_:_6+-y-=i  (-24+25)=|- j 
whence  a; - 1  =  ±\^  and  ;r  =  |  +  ^  =  |  =  3,  oric=|-|  =  |  =  2. 

Ex.  2.  x^-  X  ^l.     Here  x''  -x  +  ^(4)'  =  f  +  ]-  =  1  T 
whence  x  '-\-  ±\  and  .i'  =  -^  +  1  =  1^,  or  x-  \-\  =  -\. 


lOS  'QtJADItA  ^IC   EQUATIONS. 

Ex.  56, 

1.  aj^  +  7a;  =  8.  2.  x"  -  Ux  =  G8.  3.  x'  +  25x  =  -  100. 

4.  x''  +  13a;  =  -  12.    5.  x^  +  19a;  =  20.  G.  a;^  +  Ilia;  =  3400. 


126.  If  the  coefficient  of  a?  be  a  fraction,  its  half  will, 
of  course,  be  found  by  halving  the  numerator,  if  pos- 
sible— if  not,  by  doubling  the  denominator. 

Ex.  1.  X'  +  \'-x  =  19.     Here  x''  +  -'^x  +  (^y  =  19  +  -V-  =  ^^ ; 
whence  a;  +  |  =  ±  Y",  ^^^  x  = -^  +  -^^-=^,or  x  =  -  ^-  Y:  =  -  ^j- 

Ex.  2.  x'+  y^x  =  74.  Here  x''^  ^^-x  +  {\iy  =  74  +  i«|  =  i^  ; 
whence  x  +  U=  ±fj,  and  x~  -  T|+fJ=7f ,  or  a;=  -  fj  - 14  =  -  l^* 

Ex.  67. 
1.  a;''-^a;=34.  2.  a;^-fa;  =  27.  3.  x'  +  Ja;  =  8G. 

4.  a;^  -  -2y«a;  =  144.  5.  a;^  +  yV^  =  145.        G.  a;'^  -  f  f  a;  =  147. 


127.  In  the  following  Examples  the  equations  M-ill 
first  require  reduction ;  and  since  theKule  requires  that 
the  coeff.  of  a?'  shall  be  +  1,  if  it  have  any  other  coefF., 
we  must  first  divide  each  term  of  the  equation  by  it. 

Ex.  3a;^  -  20a;  =  5.  Here  a;'-  -%^-x  =  |,  and  x^-  -^/x  +l^  =  i^; 
whence  x=  J  (10  ±  VH-^)?  th*  roots  being  here  surd  quantities. 

Ex.  58. 

1.  a;  =  5  +  ^x\        2.  2a;  =  4  +  - .      3.  -jl-a;^  -  |a;=  ^V  (H^  +  18)- 
4.  lla;'-9a;=llj.        5.  J  (a;^-3)  =i  (a;-3).     C.  2a;''  +  l=ll  (a;  +  2). 

7.  a;  -  -y— -  =2.  8.  ^  +  ^ +  ^— ^  =  0. 

a;^  +  5  o      o  +  a;     o  +  2a; 

_   a;  +  22     4_9.r-6  a;  +  2     4-a; 

^•~3~~~^-~2~'     ,        ^^'  ^i-  ~2r-^^' 

12_      _4_  _  _32_  jr        x_+_l  _  13 

*5-a;     4:  -x~  X  +  2'  'a;  +  l         x     ~6* 


128.  An  equation  of  the  form  ax^  +  hx+c  =  0,  or  ax^  +  hx  =  -c 
(where  a,  &,  c,  are  any  quantities  whatever),  may,  however,  be 
Bolved  as  follows,  without  dividing  by  the  cocflBcient  of  a*'. 


QUADRATIC   EQUATIONS.  103 

Multiply  every  term  hjAa,  and  add  h^  to  each  side  ; 

then  Aa^x^  +  4ahx  +  1)^  =  h^-  Aac.  whence  x  =  ■ zz . 

2a 

Ex.  1.  2a;'-7aj+3=0,  or  2x^-7x  =  -  3.  Here,  mult,  by  4x2=8, 
and  add  7'  =  49  to  each  side ;  then  IGoj^  -  6(jx  +  40  =  49  -24  =25  ; 
.-.  4;r  -  7  =  ±  5,  and  a;  =  J  (7  ±  5)  =  3  or  |. 

The  advanced  student  will  find  it  well  to  accustom  himself  to 
apply  at  once  (by  memory)  the  formula  above  obtained  for  x, 

Ex.  2.  (3.r  -  2)  (1  -  x)  =  4,  or  ^x^-5x  +  (j  =  0. 

Here  x=^  (6  ±  V25^72)=^  {5±  ^-41},  the  roots  being  impossible. 

Ex.  59. 

^    _1 1__J^  2      48    ^    165        g 

'x-1      a;  +  3~35'  'a;  +  3~a;  +  10 

^    x  +  4     7-x     4.r  +  7     ,  .    Sa;-7     4a; -10     ^, 

3.  —5 =  — -r 1.  4. +  ■ ^  =  31 

3a;-3         9  x  x  +  5         ^ 

^      2x        2x-5      ^,  ^    2x  +  9     4x-Z     ^    3a;-16 

5.    5   + ?.-   =-8^  O.    Pi +  -z =  6  + r-^r • 

a;-4ic-3         ^  9  4a; +  3  18 

^      6x        3a;  -  2  4a;  +  7      5  -  a;  _  4a; 

"•  ^74"  2:^33  =  ^*  "l9~  ■"  3T5  "  T' 


129.  The  Yootsofx'+px+q=:0aYe{12S)-^2^±^ip'-q : 
hence,  (i)  ifip^>q,  we  shall  have  j^j/  -q positive^  and 
.'.  '^^i^'  -q  Vi possible  quantity :  and  since,  in  one  root, 
it  is'  taken  with  +,  and  in  the  other  with  -,  the  two 
roots  will  be  7'eal  and  different  in  value ; 

(ii)  if  i^/  =  2',  we  shall  have  ii/-y=0,  and,  there- 
fore, the  two  roots  will  be  Q'eal  and  equal  in  value. 

(iii)  if  ^p^  <  q^  w^e  shall  have  \p^  -q  negative^  and 
V\p'-q  impossible^  and  so  the  two  roots  Avill  be  im- 
possible. 

Hence,  if  any  equation  be  expressed  in  the  form 
x'+px-^-q^O^  its  roots  w^ill  be  real  and  different^  real 
and  equals  ov  impossible^  according  as^^>,  =,  or<4(^. 

So  also  in  the  more  general  equation,  ax^+hx+c^O^ 
the  roots  w^ll  be  real  and  different^  real  and  equals  or 
impossible^  according  as  J"  >,  ==,  or  <4(X<?. 


104  QUADRATIC   EQUATIONS. 

130.  If  a,  /3  represent  the  two  roots  of  x^-\-j?x-\-q=:0^ 
then  -p  =1  a+  ^y  and  q  =  a/3. 


For     a  =  -i-p+  Vip'  -q,     iS  =  - Ji>- '^jy-^; 
.-.  a+^  =  -p,  and  a^  =  \p^-  (ip"-q)  =  q^ 

Hence,  when  any  quadratic  is  reduced  to  the  form 
x^  +px  -\-  q  =z  0,  vre  have 
coeff.  of  2"^^  term,  with  sign  changed,  =5w^;z  of  roots. 

and  S""*^  term  =product  of  roots. 

ThuSj  in  (124),  the  equation,  when  expressed  in  this  form,  is 
x"^  -  Gx-7=0,  and  the  roots  are  there  found,  7  and  -1 ;  and  here 
+  0=7+  (-1)  -  sum  of  roots,  and  -7=7  x  (-1)  -.product  of  roots. 

So  also  ax^-\-hx+c=^0^  expressed  in  this  form,  becomes 

"b  G  1)  C  ' 

x^+  ~x-\-  -  =  0'  .•.--=  Slim  of  roots,  -  =  vroducL 
a        a  a  a 

131.  If  a^  phe  the  roots  of  x^  +px  +  q  —  0^  then 

x"  +px  ■\-  q^ix-a)  {x-^). 
For,  (130)  x^+px  +  qz=x'-{a  +  ^)x-\-  a/S 

=  x''-ax  ^'I3x  -f  a^={x-a)  {x-/3). 

So  also  if  a,  /She  the  roots  oi*ax^-rhx-\-c=Oj  we  have 
ax^  +  hx  +  c  =  a  ix''  +  -  x+  -j  =  a{x-a)  {x- ^). 

132.  Hence  Ave  may  form  an  equation  with  any 
given  roots. 

Thus  with  roots  2  and  3,  we  have  (;r-2)  (rr-3)  =a;'-5j  +  G=0  ; 
with  roots  -  2  and  ^,  we  have  {x  +  2)  (aj  -  J)  =  a;*  +  Ja;  -  ^  =  0,  or, 
clearing  it  of  fractions,  Ax^  +  7a;  -  2  =  0. 

This  law  is  not  confined  to  quadratics,  but  may  be 
shewn  to  be  true  for  equations  of  all  dimensions. 
Thus  the  biquadratic  whose  roots  are  - 1,  2,  -  2,  3,  is 
(a;  +  1)  (a;  -  2)  {x  +  2)  (a;  -  3)  =  x'  -  2x^  -  Ix"  +  8ar  +  12  :-  0. 


QUADRATIC  EQUATIONS.  105 

133.  If  one  of  tlie  roots  be  0,  the  corresponding 
factor  will  be  ^  -  0  or  x. 

Thus,  with  roots  0,  1,  3,  wc  have  x{x-V)  {x-Z)=x^-\x^ ^■Zx-^, 
In  such  a  case  then  x  will  occur  in  every  term  of  the 
equation,  and  may  therefore  be  struck  out  of  each  ;  but 
let  it  be  noticed  that,  whenever  we  thus  strike  an  x  out 
of  every  term  of  an  equation,  it  must  not  be  neglected, 
since  such  an  equation,  as  it  originally  stood,  would  be 
satisfied  by  fi?=0,  which  is  therefore  one  of  its  roots. 

ThuSj  in  the  ahove  equation,  we  may  strike  an  x  out  of  every 
terra,  and  thus  reduce  it  to  a?*  -  4ic  +  3  =  0,  which  gives  us  the  two 
roots,  1  and  3  ]  but,  besides  these,  we  have  the  root  ic  =  0. 
Ex.  60. 

Form  the  equation  with  roots 
1.  7  and  -  3,  2.  f  and  -  #.  3.  3,  -  3,  f,  -  \, 

4.  0,  1,  2,  3.  5.  0,  -  i,  li,  -  1.        G.  0,  -  1,  2,  -  2,  \. 


We  shall  now  give  a  few  examples  of  quadratic 
equations  of  two  unknowns.  The  solution  of  these  is 
generally  more  difhcult :  but  there  are  three  cases  of 
frequent  occurrence,  for  which  the  following  observa- 
tions will  be  useful. 

134.  (i)  Express,  when  possible,  by  means  of  one  of 
the  equations,  either  of  the  unknowns  in  terms  of  the' 
other,  and  put  this  value  for  it  in  the  other  equation. 


PI  1      2^H-y^ 

Ex.  1.  ic  +  -  =  — o— 

X  ^  y  _\x-y 

__  _  ______ 


(i) 


(ii) 


Prom  (i)  we  get  y-x^\  ;  and,  putting  this  value  for  y  in  (ii),  wo 
have =  -~^,  whence  x-1  or  -  1  and  .-.  2/=a;+l=3  or  f. 

The  given  equations  have,  therefore,  two  pairs  of  roots, 
a?  =  2  and  y  =  3,     or  a*  =  -  ^  and  y  -l» 


106  QUADRATIC   EQUATIONS. 

135.  (ii)  When  either  of  the  two  equations  is  homo- 
geneous with  respect  to  x  and  y,  in  all  those  terms  of  it 
w^iicli  involve  x  and  ?/,  put  y  =  vx,  by  which  means 
we  may  generally  without  difficulty  obtain  an  equa- 
tion involvii^g  V  only,  which  being  determined,  x  and 
y  may  then  be  found. 

Es.  2.  x''  +  xy  +  y^  =  7)     (i) 

2x+  Zy  =  s\     (ii) 
Here  putting  vx  for  y^  x'  (1  +  v  +  v^)  =  7,         («) 
x(2^  3r)  =  8  ;        (/3) 
•.  dividing  (a)  by  the  square  of  (/3\  the  x"^  disappears,  and  we  have 
l+i)  +  ?)^      7      ^  ^      ,^ 

-(27T.)"  =  64'''^"°''^^^^"^'1^5 

and  from  (3),  ic  (2  +  G)  =  8,  or  aj  =  1,  and  y  =  ra;  =  2, 
or  X  (2  +  54)  =  8,  or  x  --=  -J,  and  y  =  vx=  2*, 

(iii)  When  each  of  the  two  equations  is  symmetrical 
with  respect  to  x  and  ?/,  put  ?^+?;  for  x  and  '^/-t'  for  y. 

Def.  An  expression  is  said  to  be  symmetrical  with  respect  to  x 
and  y,  when  these  quantities  are  similarly  involved  in  it :  thus 

x^  +  x-y^  +  y\      4xy  +  5^  +  5y  -  1,      2x^  -  Zx^'y  -  Zxi/  +  2y\ 
are  symmetrical  with  respect  to  x  and  y. 
Ex.3.  ai" +  y^=lSxy-)     (i) 

OJ   +  7/   =  12      \      (ii) 
Put  '?i  +  V  for  0^,  and  ?/  —  v  for  ?/ ; 

then  (!)  becomes  (a  +  vy  +  (u  -  i^y  =  18  (u  +  v)  (u  -  -c), 

or  u''  +  3?^y^  =  9  (w^ -?)-);         (a) 
and  (ii)  becomes  (tc  +  ^c)  +  (u  -v)  ~  12,  whence  u  =  0  ] 
putting  this  for  u  in  (a),  210  +  18y'  =  9  (36  -  v'),  whence  v  =  ±  2 ; 
,'._x  =  u  +  V  =  C)  ±  2=  8  or  4,  and  y  =  'w-'y=G±2  =  4or8. 
13C.  The  preceding  are  general  methods  for  the  solution  of 
equations  of  the  kinds  here  referred  to,  and  will  sometimes  succeed 
also  in  other  equations  ;  yet  in  many  of  these  cases  a  little  inge- 
nuity will  often  suggest  some  step  or  artifice,  by  which  the  roots 
may  be  found  more  simply,  but  for  which  no  rules  can  be  given. 

The  methods  pursued  in  the  two  following  examples  are  worthy 
.of  notice  in  this  respoct. 


QUADRATIC   EQUATIONS.  107 

Ex.  4.  Zx"  -  2xy  =  15 )      (i) 

2a;  +    3y  =  12  ^     (ii) 
Mult,   (i)  by  3,  9a;-  -  Q>xy  =  45, 

....    (ii)  by  2a;,  4a;'  +  (Sxy  =  24a; ; 

.'.  adding,  13a;*=45  +  24a;,  or  13a;'-24a;=45,  whence  a;=3  or  -1^. 
and  from  (ii)     y  =  \  (12-2a!)=2  or  4||.* 
Ex.  5.  a;'  +  y''  =  25  ;       (i) 

2xy=2A\      (ii) 
Here  adding,  x^  +  2xy  +  y^  =  49,  whence  x  +  y  =  ±7 1 
subtracting,  x^  -  2xy  +  y^  =    1,  whence  a;  -  y  =  ±  1 : 


and  X  -y 


=  +  7)  x+  y  =  +  7  ) 

=  +  IS  x-y  =  -1 j 


.*.  2a;  =  8,  and  a;  =  4,  2x  =  0.  and  a;  =  3, 

2y  =  6,  and  y  =  3;  2y  =  S,  and  y  =  4 : 

similarly,  by  combining  the  equation  x  +  y  =  —7  with  each  of  the 
two  x-y=±l,  we  should  get  the  other  two  pairs  of  roots 
a;  =  -  4,  2/  =  -  3,  and  a;  =  -  3,  y  =  -  4. 

Ex.  61. 

1.  i^{^x-^5y)-vl(4.x-Zy)=:^n    2.  a;^  +  2/'=25)    3.  x'^.y''=2^l 

3a;=  + 22/^=179^  x+y=  1)       4^  +  3a;=:24^ 

4.  2  (a;-?/)  =11 )  5.  a;'  +  a'2/=66)  6.  a;-2/=2  ) 

a'2/=20  \  X'-y''=ll )  15  (a;^-2/-)=lGa'2/  S 

^  aj*-*  _  85     4a;  1  8.  a;?/=(a;-^)(y  +  f) )  9.        a'+y=G) 

'  V  "  ¥  "  y  I        2;-2/'=  (a;'  +  3)  {y-^4)  \  ^3  +  2/'=72  \ 

~     x-y^2     ] 

10.  3a;y  +  2a;  +  y  =  485  )      11.  a;-y=    1)        12.  a;^  +  «/*  =  189  ) 

3a;  =  2y  )  a;^-  2/'=  19  \  xhj  +  xy""  =  180  S 

13.  x-^y=a  )     14.     a;?/=aM      15.  V'^+V^=^?       1^-  ic'  +  ^'2/=«'; 

aj*+2/^=5^^  x-y-h  f  a^+2/=9)  y'  +  a;y=&^^ 

137.  In  the  solution  of  Problems,  depending  on  quadratic  and 
higher  equations,  there  may  be  two  or  more  values  of  the  root,  and 
these  may  be  real  quantities,  or  impossible.  In  the  former  case, 
we  must  consider  if  any  of  the  roots  are  excluded  by  the  nature 
of  the  question,  which  may  altogether  reject  fractional^  or  nega- 
tive, or  surd  answers:  in  the  latter  case,  we  conclude  that  the 
solution  of  the  proposed  question  is  arithmetically  impossible. 


108  QUADRATIC   EQUATIONS. 

Ex.  1.  What  number^  when  added  to  30,  will  le  less  than  its 
square  hy  12  1 

Let  X  be  the  number  ;  then  30  -^  x  =  x'^  ~\2.  whence  a;  =  7.  or 
-  6  :  and  here  the  latter  root  would  be  excluded,  if  we  require 
only  positive  numbers. 

Ex.  2.  A  pefson  bought  a  number  of  oxen  for  £120;  if  he  had 
bought  3  more  for  the  same  money  ^  he  would  have  paid  £2  less  for 
each.    Hoio  many  did  he  buy  f 

Let  X  be  the  number  he  bought :  then  the  price  actually  given 

120                 120       120 
for  each  was ,  and  .*. ^  = 2,  whence  x  -  12,  or  -  15, 

X  X  ^Z        X 

which  latter  root  is  rejected  by  the  nature  of  the  Problem. 

Ex.  3.  The  sum  of  the  squares  of  the  digits  of  a  number  of  two 
places  is  25,  and  the  product  of  the  digits  is  12.     Mnd  thejiumber. 

Let  X.  y  be  the  digits,  so  that  the  number  will  be  lOa;  +  y ;  then 
x-+y^=  25,  and  xy  =  12,  from  which  equations  we  get  x  =  o.  y  -  4^ 
or  a?  =  4, 2/  =  3,  and  the  number  will  be  34  or  43.  In  this  case 
both  the  roots  give  solutions. 

Ex.  4.  Mnd  two  numbers  sucJi,  that  their  sum,  2^roductj  and 
difference  of  their  squares  may  be  all  equal. 

Hero  assume  x  +  y  and  x-yfor  the  two  numbers :  [this  step 
should  be  noticed,  as  it  simplifies  much  the  solution  of  problems  of 
this  kind  :]  then  their  sum  =  2x,  their  product  =  ic'  -  y^^  and  the 
difference  of  their  squares  =  4r?/ ;  .*.  (i)  2x  =  4.ry,  (ii)  2x  =  x^-  y'^ ; 
from  (i)  y  =  ^,  from  (ii)  2x  =:  x' -  \^  whence  aj  =  |(2  ±  ^5) ;  and 
.-.  a;  +  2/  =  1  (3  ±  ^5),  x-y  =  \(\±  ^5),  the  numbers  required. 

Ex.  5.  Find  two  numbers  ichose  difference  is  10,  and  2>roduct 
one-third  of  the  square  of  their  sum. 

Let  X  =r.  the  bast,  and  .t  +  10  =  the  greater ;  then  x(x+  10) 
=  J  (2x+10y,  wlionce  a;=-5±5V-3,  which  are  impossible.  The 
question  in  fiict  amounts  to  asking  for  two  numbers  x  and  y,  such 
that  xy  =  i(x  +  yy^  or  Sxy  =  x'  +  2xy  +  t/^,  or  xy  =  a;'^  +  t/',  which 
may  be  easily  shewn  to  be  impossible  :  for  (x  -  yy,  or  x^-  2xy  +  ?/^ 
is  necessarily  positive  (being  a  square  quantity)  whatever  x  and  y 
may  be,  and  .-.  ar  f  y"  must  be  greater  than  2xy. 


QUADRATIC   EQUATIONS.  109 

Ex.  62. 

1.  There  are  two  numbers,  one  of  which  l<3  f  of  the  other,  and 
the  difference  of  their  squares  is  81 :  find  them. 

2.  The  difference  of  two  numbers  is  |  of  the  greater,  and  the 
sum  of  their  squares  is  35G  :  find  them. 

3.  There  are  two  numbers,  one  of  which  is  triple  of  the  other, 
and  the  difference  of  their  squares  is  128  :  find  them. 

4.  In  a  certain  court  there  are  two  square  grass-plots,  a  side  of 
one  of  which  is  10  yards  longer  than  a  side  of  the  other,  and  the 
area  of  the  latter  is  -^j  of  that  of  the  former.  "What  are  tho 
lengths  of  the  sides  ? 

5.  What  two  numbers  make  up  14,  so  that  the  quotient  of  the 
less  divided  by  the  greater  is  ^V  of  the  quotient  of  the  greater 
divided  by  the  less  ? 

6.  A  draper  bought  a  piece  of  silk  for  £16  4s,  and  the  number 
of  shillings  which  he  paid  per  yard  was  ^  the  number  of  yards. 
How  much  did  he  buy  ? 

7.  A  detachment  from  an  army  was  marching  in  regular  column, 
with  5  men  more  in  depth  than  in  front ;  but  on  the  enemy  com- 
ing in  sight,  the  front  was  increased  by  845  men,  and  the  whole 
was  thus  drawn  up  in  5  lines :  find  the  number  of  men. 

8.  What  number  is  that,  the  sum  of  whose  third  and  fourth 
parts  is  less  by  2  than  the  square  of  its  sixth  part  ? 

9.  There  is  a  number  such  that  the  product  of  the  numbers 
obtained  by  adding  3  and  5  to  it  respectively  is  less  by  1  than  the 
square  of  its  double  :  find  it. 

10.  There  is  a  rectangular  field,  whose  length  exceeds  its 
breadth  by  16  yards,  and  it  contains  960  square  yards  :  find  its 
dimensions. 

11.  The  difference  between  the  hypothenuse  and  two  sides  of 
a  right-angled  triangle  is  3  and  6  respectively  :  find  the  sides. 

12.  What  two  numbers  are  those  whose  difibrence  is  5,  and 
their  sum  multiplied  by  the  greater  228  ? 

13.  A  labourer  dug  two  trenches,  one  6  yards  longer  than  the 
other,  for  £17  I65,  and  the  digging  of  each  cost  as  many  shillings 
per  yard,  as  there  were  yards  in  its  length  :  find  the  length  of  each. 

14.  The  plate  of  a  looking-glass  is  18  inches  hy  12,  and  it  is  to 
be  framed  with  a  frame  of  uniform  width,  whose  area  is  to  be 
equal  to  that  of  the  glass  :  find  the  width  of  the  frame  ? 


110  QUADRATIC   EQUATIONS. 

15.  There  are  two  square  buildings,  paved  with  stones,  each  a 
foot  square.  The  side  of  one  building  exceeds  that  of  the  other 
by  12  feet,  and  the  two  pavements  together  contain  2120  stones: 
find  tlie  sides  of  the  buildings. 

IG.  A  person  bought  a  certain  number  of  oxen  for  £240,  and, 
after  losing  3,  sold  the  rest  for  £8  a  head  more  than  they  cost 
him,  thus  gaining  £59  by  the  bargain :  what  number  did  he  buy  ? 

17.  A  tailor  bought  a  piece  of  cloth  for  £147,  from  which  he  cut 
off  12  yards  for  his  own  use,  and  sold  the  remainder  for  £120  5«, 
charging  5  shillings  'per  yard  more  than  he  gave  for  it.  Find  how 
many  yards  there  were,  and  what  it  cost  him  per  yard. 

18.  The  fore-wheel  of  a  carriage  makes  6  revolutions  more  than 
the  hind-wheel  in  going  120  yards ;  but  if  the  circumference  of 
each  were  increased  by  3  feet,  the  fore-wheel  would  make  only 
4  revolutions  more  than  the  hind  one  in  the  same  space.  What 
is  the  circumference  of  each  ? 

19.  By  selling  a  horse  for  £24,  I  lose  as  much  per  cent,  as  it 
copt  me.     What  was  the  prime  cost  of  it  ? 

20.  Bought  two  flocks  of  sheep  for  £15,  in  one  of  which  there 
were  5  more  than  in  the  other  ;  each  sheep  in  each  flock  cost  as 
many  shillings  as  there  were  sheep  in  the  other  flock.  How  many 
were  there  in  each  ? 

21.  A  and  B  take  shares  in  a  concern  to  the  amount  altogether 
of  £500 :  they  sell  out  at  p^r,  A  at  the  end  of  2  years,  B  of  8,  and 
each  receives  in  capital  and  profit  £297.  How  much  did  each 
embark  ? 

22.  A  and  B  distribute  £5  each  in  charity :  A  relieves  5  persons 
more  than  i>,  and  B  gives  to  each  \8  more  than  A,  How  many 
did  they  each  relieve  ? 

23.  There  is  a  number  of  three  digits,  of  which  the  last  is  double 
of  the  first :  when  the  number  is  divided  by  the  sum  of  the  digits, 
the  quotient  is  22;  and,  when  by  the  product  of  the  last  two,  11. 
Find  the  number. 

24.  Find  three  numbers,  such  that  if  the  first  be  multiplied  by 
the  sum  of  the  second  and  third,  the  second  by  the  sum  of  the 
first  and  third,  and  the  third  by  the  sum  of  the  first  and  second, 
the  products  shall  be  2G,  50,  and  5G. 


INDETERMINATE   EQUATIONS.  Ill 

We  have  seen  that  when  we  have  only  one  equa- 
tion between  two  unknowns,  the  number  of  solutions 
is  unlimited^  and  the  equation  is  indeterminate.  We 
shall  here  make  a  few  remarks  upon  the  simpler  kinds 
of  such  equations. 

138.  If  one  solution  be  given  of  the  equation 
ax  ±,hy  =^  c^  all  the  others  may  be  easily  found. 

For  let  aj==a,  2/=/3,  be  one  solution  of  the  equation 
ax  +  ly—G  ;  then  ax +hy  =^  c  =^  aa-^-  h^^  or  a  (x—a) 
+  ^  (2/~yS)=0,  which  equation  is  satisfied  by  x-a=  -  ht, 
y-^=iat,  where  t  may  be  any  quantity  whatever,  pos- 
itive or  negative.  Hence  the  general  values  of  x  and  y 
are  given  by  the  expressions  x  =  a-U,  y  =z  ^ -\-  at 

If  the  given  equation  be  of  the  form  ax  -  h/=c,  we 
should  obtain  in  the  same  way,  x=a  +  It^  y=:^-\-  at, 
the  same  as  w^e  get  by  writing  ~  h  for  i  in  the  above. 

If  we  require  only  integral  values  of  a?  and  y,  the  n"* 
of  solutions  will  be  limited ;  i\\Q  above  results  will 
still  apply,  only  we  must  now  have  a,  y8,  t  all  integers. 

139.  It  may  be  shewn  however  that  there  can  be 
no  integral  solution  oi  ax±  'by=c,  if  a  and  Shave  any 
conimon  factor,  not  common  also  to  c. 

For  let  a=^mdy  h^nd,  while  g  does  not  contain  d ; 

thenmdx±7idy=c,ovmx±ny=:z  -3=a  fraction,  whichis, 

of  course,  impossible  for  any  integral  values  of  x  and  y. 
We  shall  suppose  then  in  future  that  a  \s>  prime  to  5. 

140.  To  solve  the  equation  ax^hj  ^^  c  m  integers. 
If  w^e  can  discern  one  solution,  we  may  apply  (138). 

Thus  13a;  -  9^  =  17  is  satisfied  by  a;  =  2,  y  =  1 ; 
whence  ISaj-Oy  =  17  =  13  x  2-9  x  1,  or  13  (aj-2)  =  9  (y-l), 
which  is  satisfied  by  a;  -  2  =  9^,  y  -  1  ^  13^,  so  that  the  solution 
is  ic  =  2  +  9#j  y  =  1  +  13^,  where  t  may  have  any  integral  value. 


112  INDETEEMINATE   EQUATIONS. 

But  the  following  examples  will  shew  the  simplest 
general  method  of  solving  such  an  equation. 

Ex.  1.  Find  the  integral  solutions  of  ^x  +  5y  =  73. 

Divide  by  the  lowest  coefficient,  and  express  the  improper  frac- 
tions wliich  niay  arise  as  mixed  numbers ; 

then  a;  +  y  +  f  2/  =  24  +  J,  or  a;  +  y  -  24  =  J  -  |y  =  — ^ — . 

l-2v 
Now.  since  a;  +  y  -  24  is  integral,  so  also  is  — —- ,  and   any 

o 

multiple  of  it ;  multiply  it  then  by  such  a  number  as  will  make  the 

coeff.  of  y  din,  hy  the  derC  with  rem''  1,  i.  e.  in  this  case,  mult,  it  by  2 ; 

2 -All       2  -  V         .  2  -  V  .    . 

then  — :j--  or  — ^  -  y  is  int.,     /.   — ~-  is  mt.  =  t  suppose  ; 

hence  2  -  2/ =  3«,  or  7/ =2 -3^,  and  a;  =  J  (73 -5?/)  =  21  +  5f. 

Thus,  if  we  take  t  =  0,  then  aj  =  21,  y  =  2  j 
if  ^  =  1,  aj  =  26,  1/  =  -  1 ;  if «  =  - 1,  ic  =  16,  y  =  5  ;  &c. 

If  we  require  only  positive  integral  values  of  x  and  y,  then  we 
caanot  take  t  positively  >  f,  nor  therefore  >0,  or  negatively  >  -^/, 
nor  therefore  >  4 ;  hence  the  values  for  t  range  from  -  4  to  0  in- 
clusively, and  thus  there  will  be  only  5  positive'miQgv^X  solutions. 

N.  B.  It  may  be  shewn  that  it  is  always  possible  to  find  such  a 
number  for  multiplier  as  we  have  employed  above,  which  shall 
be  less  than  the  denominator :  and  this  is  the  reason  why  we  divide 
by  the  least  of  the  two  coefficients,  in  order  to  have  the  multiplier 
as  low  as  possible.  But  when  the  denominators  are  both  large, 
a  little  ingenuity  will  save  the  trouble  of  searching  for  such  a 
number,  by  some  such  reasoning  as  that  in  the  next  Ex.,  it  being 
noticed,  that  the  point  to  be  aimed  at  is,  to  get  the  coefficient  of  y 
(or  of  a;,  as  the  cas^  may  be)  in  the  numerator  to  be  unity, 

Ex.  2.  Solve  in  positive  integers  39a;  —  5Gy  =  11. 

1,        ,,  i7y +  11  .     .  ^         ,        342/ +  22 

Here  x-y-l'^y  =  l^]    .-.  — ^ —  is  mt,  and  .-.  —^ — , 

342/  +  22     by-22       ^       40^-176  ,     y  -  20 

and  .-.  y  -  -^  or-^-,  and  .-.  -- —  or  2/  -  4  +  ^-3^-  ; 

7/-20 
let  ^^^^  =  ^;  .-.  2/  =  39i  +  20,  and  x  =  3V  (U  +  56?/)  =  56^  +  29. 

If  we  take  ^  =  0,  then  a;  =  29,  2/  =  20,  which  arc  the  least  positive 
integral  values  they  admit  of:  but  the  number  of  such  values  is 
here  unlimited^  since  we  may  take  a?iy  positive  value  for  t. 


INDETERMINATE   EQUATIONS.  113 

Ex.  3.  Find  the  least  number  which  when  divided  by  14  and  5 
will  leave  remainders  1  and  3  respectively. 
Let  the  number  required  I^=l^x+l=6i/  +  ^ ;  then  14a;-52/=2, 

and  here  2x+^x-i/=^j  or  2x-y=  — ^ —  ;  hence  — ^—  is  integral, 

and  .'.  also  — - — -.  and  —z — ,  which  put  =^  t: 
0  o      . 

whence  x  =  Z  -  5t,  fxnd  y  =  ^  (Ux  -  2)  =  S  -  UL 

If  we  take  t  -  0,  we  have  a;  =  3,  y  =  8,  which  arc  the  least 
positive  integral  values  they  admit  of,  and  therefore  the  least 
value  of  iVis  14.3  +  1  =  5.8  +  3  =  43 ;  but  the  n°  oi positive  values 
is  unlimited,  since  we  may  take  any  negative  value  for  t. 

N.B.  It  appears  from  Ex.  1,  2,  3,  that  when  only  positive  integral 
solutions  are  required,  the  n*  of  them  will  be  limited  or  not,  ac- 
cording as  the  equation  is  of  the  form  ax  +  hy  =c^  or  ax  -hy  =  c, 

Ex.  4.  Find  the  least  integer  which  is  divisible  by  2,  3,  4,  with 
remainders  1,  2,  3. 

Let  ]^=2x  +  l  =  32/+2=42J  +  3:  then  (i)  2a;  -  3y  =  1,  whence,  as 
before,  a;=3^-l,  7/r=2^-l ;  and  (ii)  2a;-4^=2,  or  3^-2^=2,  whence 
t=2t\  2=3^'-l :  .-.  aj=6i5'-l,  y=W-\,  s=3^ -1,  whence,  putting 
t'  =  1,  we  get  a?  -  5,  and  JSf-  2a;  +  1  =  11. 

Ex.  5.  In  how  many  ways  may  £80  be  paid  in  £s  and  guineas  ? 

Let  a;  =  n'*  of  £s,  y  =  n**  of  guineas  ;  then  20a;  +  2\y  =  n"  of 
shillings  in  £80=1000,  and  x-\-y+^^y=%0:  put  2^2/=^ 5  .\y=20t^ 
and  X  =  2V  (1000  -  2ly)  =  80  -  21^,  which  gives /bi^r  solutions,  or 
rather  three^  if  we  omit  the  solution  t  --  0,  which  gives  y  =  0. 

[In  the  Answers  we  shall  omit  all  zerO'Valiies  for  x  or  y.] 
Ex.63. 

1.  Find  the  positive  integral  solutions  of 

2a;  +  3y  =  9,  ^x  +  29?/  =  150,  3a;  +  29y  =  151,  Ix  +  15?/  =  225. 

2.  Find  the  least  positive  integral  solution  of 

19a;  -  Uy  =  11,  17a;  =ly  +  1,  23a;  -  9y  =  929,  8a;  =  23y+l9, 

3.  Find  the  number  of  positive  integral  solutions  of 

3a;  +  4y  =  39,  8a;  +  Uy  =  500,  7x  +  Uy  =  405,  2x  +  7y=:  125. 

4.  Given  x-2y  +  z  =  o  and  2a;  +  y  -  s  =  7,  find  the  least  values 
of  a;,  y,  «,  in  positive  integers. 

5.  A  person  distributed  4s  2d  among  some  beggars,  giving  7d 
each  to  some,  and  Is  each  to  the  rest :  how  many  were  there  in  all  ? 


114  INDETERMINATE   EQUATIONS. 

6.  In  how  many  ways  could  12  guineas  be  made  up  of  lialf- 
guineas  and  half-crowns  ?  In  how  many  ways,  of  guineas  and 
crowns  ? 

7.  How  man}^  fractions  are  there  with  denominators  12  and  18j 
whose  sum  is  f  |  ? 

8.  A  wishes  to  pay  B  a  debt  of  £1  12s,  but  has  only  half- 
crowns  in  his  pocket,  while  J3  has  only  fourpenny-pieces ;  how 
may  the)'-  settle  the  matter  most  simply  between  them  ? 

9.  "What  is  the  least  number,  whichj  divided  by  3  and  5,  leaves 
remainders  2  and  3  respectively?  What  is  the  least,  which 
divided  by  3  and  7,  leaves  remainders  1  and  2  ? 

10.  A  person  buys  two  pieces  of  cloth  for  £15,  the  one  at  Ss, 
the  other  at  llsjo^r  yard,  and  each  containing  more  than  10  yards : 
how  many  yards  did  he  buy  altogether  ? 

11.  In  how  many  ways  can  £1  be  paid  in  half-crowns,  shillings, 
and  sixpences,  the  number  of  coins  used  at  each  payment  being  18  ? 

12.  A  person  counting  a  basket  of  eggs,  which  he  knows  are 
between  50  and  GO,  finds  that  when  he  counts  them  3  at  a  time 
there  are  2  over,  but  when  he  counts  them  5  at  a  time,  there  are 
4  over  :  how  many  were  there  in  all  ? 

13.  If  I  have  9  half-guineas  and  G  half-crowns  in  my  purse,  how 
may  I  pay  a  debt  of  £4  Ils6d7 

14.  A  person  in  exchange  for  a  certain  number  of  pieces  of 
foreign  gold,  valued  at  29s  each,  received  a  certain  number  of  sover- 
eigns under  fifty,  and  Is  over  :  what  was  the  sum  he  received  ? 

15.  A  French  loicis  contains  20  francs,  of  which  25  make  £1 : 
how  can  I  pay  at  a  shop  a  bill  of  45/?*  most  simply,  by  paying  Eng. 
and  receiving  Fr.  gold  only  ?    Shew  that  I  cannot  pay  a  debt  of  455. 

16.  A  person  bought  40  animals,  consisting  of  calves,  pigs,  and 
geese,  for  £40;  the  calves  cost  him  £5  a  piece,  the  pigs  £1,  and 
the  geese  a  crown  :  how  many  did  he  buy  of  each  ? 

17.  Find  the  least  integer  whi«h  when  divided  by  7,  8,  9, 
respectively,  shall  leave  remainders  G,  7,  8. 

18.  Three  chickens  and  one  duck  sold  for  as  much  as  two 
geese ;  and  one  chicken,  two  ducks,  and  three  geese  were  sold 
together  for  25s  :  what  was  the  price  of  each  ? 

19.  Find  the  least  odd  number  which  when  divided  by  3,  5,  7, 
shall  leave  remainders  2,  4,  G. 

20.  Find  the  least  multiple  of  7,  which  divided  by  2,  3,  4,  5,  6, 
leaves  always  u?iit7/  for  remainder. 


CHAPTER  X. 

ARITHMETICAL,    GEOMETKICAL,    AND   HARMONICAI. 
PROGRESSION. 

141.  Quantities  are  said  to  be  in  Arithmetical  Pro- 
gression^ when  tliey  proceed  by  a  common  difference. 

Thus,  1,  3,  5,  7j  &c.,  8,  4,  0,  -4,  &€.,  a^  a  +  d^  a  +  2d^  a  +  M,  &c., 
are  in  a.  p.,  the  common  differences  being  2,  -4,  d.  respectively, 
which  are  found  by  suhtj^acting  any  term  from  the  term  following, 

142.  Given  a  the  first  term,  and  d  the  common  dif- 
ference of  an  AR.  series^  to  find  1  the  n"^  term^  and  S 
the  sum  of  n  terms. 

Here  the  series  will  be  a,  a+cZ,  «+2cZ,  a+Sd,  &c.y 
where  the  coeft".  of  d  in  any  term  is  just  less  hij  one 
than  the  No.  of  the  term :  thus  in  the  2"*^  term  we 
have  d^  i.  e,  Id^  in  the  o'^^  ^d^  in  the  4*^,  ScZ,  &c.,  and 
so  in  the  n^^  term  we  shall .  have  {71  -l)d\  hence 
^=^  +  (^1-1)  d, 

Ag2i\xiS=a+{a+d)+{a+'ld)+&Q,+{l-2d)+{l-d)+l, 
and  also  S=l  +  {l-d)  +  {I -2d)  +  &c.  +  {a  +  2d)-\- 
{a  +  d)-\-  a\ 
.-.  2S=:{a-tl)'\-(^a-\-l)-^{a+l)+&Q.=:{a+l)  n] 

n  n 

.'.  S=  {a-\-l)-=\2a+{n-l)d\-,  since?  =  a-\- 

2  .        ^ 

{n-l)d, 
Ex.  1.  Find  the  10*^  term  and  the  sum  of  10  terms  of  1,  5,  9,  &c. 
Here  a-1,  d  =  i,  7i  =  10  -, 

...  ^  ==  1  +  (10  -  1)  4  =  1  +  9  X  4=  37  ;  ^=  (1  +  37)  x  y>  =  190. 
Ex.  2.  Find  the  9*''  term  and  the  sum  of  9  terms  of  7,  5^  4,  &c. 
Here  a  =  7,  d=  -  ^,  71  =  9  ; 

,'.l  =  7  +  (9-1)  x''-f  =  7-8  xf  =  -5;>Sf=(7-5)xf  =  9. 
Ex.  3.  Find  the  13'^  term  of  the  series  -  48,  -  44,  -  40,  &,c. 
Here  a  =  -  48,  cZ  =  4,  ti  =  13  ; 

...  Z  =  _  48  +  (13  - 1)  4=  -  48  +  12  X  4  =  0. 


116  ARITHMETICAL,    GEOMETRICAL,    AND 

Ex.  4.  Find  the  sum  of  7  terms  of  ^  +  ^  +  ^  +  &c. 
Here  «=|,  d=  -|,  n=l ;  and  here  we  are  not  required  to  find  I : 
.'.J  using  the  second  formula,  ^5^=  (1  +  G  x  —J)  l  =  (1  -  1)  J  =  0. 
In  this  case  the  series,  continued,  is  |,  -J-,  j,  0,  -  J,  -  -J-,  -  |,  itc. 
where  the  first  7  terms  together  amount  to  zero. 

Ex.  64, 
Find  the  last  term  and  the  sum  of 
1.  2+4+G+  &c.  to  IG  terms.  2.  1  +  3  +  5+  &c.  to  20  terms. 

3.  3  +  9  +  15  +  &c.  to  11  terms.        4.  1  +  8  +  15  +  <S:c.  to  100  terms. 
5.  -5-3-1  -  &c.  to  8  terms.  G.  1+^  +  4  +  &c.  to  15  terms. 

Find  the  sum  of 
7.  |+t\  +  yV  +  <S:c.  to  21  terms.       8.  4-3-10-&C.  to  10  terms. 
9.  l+J  +  1  +  &c.  to  10  terms.        10.  i-f-V  -  &c.  to  13  terms. 
11.  i+2f+4^  +  &c.  to  20  terms.    12.  f-JJ-fi  -  &c.  to  10  terms. 


143.  By  means  of  the  equations  (i)  Z=a4-  (^2-1)  cZ, 

(ii)  S=  {ct+V)l,  and  (\{i)S=  ]2c^  +  (^^-1)  d\  %    wli^n 

2  2 

any  three  of  the  quantities  ^,  d^  Z,  n^  /S'are  given,  we 
may  find  the  others. 

We  may  also  employ  them  to  solve  many  problems 
in  A.  p.,  as  in  the  following  examples. 

Ex.  1.  The  first  term  of  an  ar.  series  is  3,  the  IS'**  term,  55; 
find  the  common  difference. 

Since  Z=55,  «=3,  7i=13,  we  have  by(i)  55=3  +  12^?,  and  .-.  rf=4^. 

Ex.  2.  What  No.  of  terms  of  the  series  10,  8,  G,  &c.  must  be 
taken  to  make  30  ?  ai^  what  No.  to  make  28  ? 

(1)  >S'=30,«  =  10,  ^  =  -  2  ;  .-.  by  (iii)  30.=  1 20  -  2  (w-l)}^; 

and  the  roots  of  this  quadratic  are  5  and  G,  either  of  which  satis- 
fies the  question,  since  the  sixth  term  of  the  series  is  zero : 

(2)  S'-=  28,  a  =  10,  d  =  -  2  ;  and  the  values  of  n  are  4  and  7, 
either  of  which  also  satisfies  the  question,  since  the  5^^  G"*,  and 
7"*  terms  of  the  series,  viz.  2.  0.  -  2,  together  =  zero. 

Ex.  3.  How  many  terms  of  the  series,  3,  5,  7,*&c.  make  up  24? 
Here  /S'=  24,  a  =  3,  d  =  2;  whence  7i  =  4  or  -G,  of  which  the 
first  only  is  admissible  by  the  conditions  of  tlie  Question. 


HARMONIC AL   PROGRESSION.  117 

Ex.  4.  Insert  3  ar.  means  between  6  and  2G. 

Here  we  have  to  find  three  numbers  between  6  and  26,  so  that 
ihajive  may  be  in  a.  p.  This  case  then  reduces  itself  to  finding  d^ 
when  0^  =  G,  l^  26,  and  n  =  b  ',  we  have  then  by  (i)  26  =  6  +  4^7, 
whence  d=  b^  and  the  means  required  are  11,  16,  21. 

Ex.  5.  The  sum  of  three  numbers  in  a.  p.  is  21,  and  the  sum 
of  their  squares,  170  ;  find  them. 

Let  a-d^  a^  a  +  d,  represent  the  three  numbers  (which  is 
often  a  convenient  assumption  in  problems  of  this  kind) ; 

then  {a-d)  +  a  -v  {a  +  d)  =  21,  and  {a  -  dy  +  a^^  {a  +  dy  =  179, 
from  which  equations  a  =  7,  d=  ±4,  and  the  Nos.  are  3,  7,  11. 
Ex.  65. 

1 .  The  first  term  of  an  ar.  series^s  2.  the  common  difference  7, 
and  the  last  term  79  ;  find  the  number  of  terms 

2.  The  sum  of  15  terms  of  an  arithmetic  series  is  600,  and  the 
common  dificrence  is  5  ;  find  the  first  term. 

3.  The  first  term  is  13  y*^,  the  common  difference  -  f ,  and  the 
last  term  §  ;  find  the  number  of  terms. 

4.  The  sum  of  11  terms  is  14^,  and  the  common  difference  is  ? ; 
find  the  first  term. 

5.  Insert  4  ar.  means  between  2  and  17,  and  4  between  2  and  -18. 

6.  Insert  9  a.  m.  between  3  and  9,  and  7  between  - 13  and  3. 

7.  Insert  10  a.  m.  between  -  7  and  114,  and  8  between  -3  and  -J. 

8.  Insert  9  a.m.  between  -2|  and  4 J,  and  9  between  -3f  and  2 J. 

9.  Find  the  3  Nos.  in  a.  p.,  whose  sum  shall  be  21,  and  the  sum 
of  the  first  and  second  =  f  that  of  the  second  and  third. 

10.  There  are  3  Nos.  in  a.  p.,  whose  sum  is  10,  and  the  product 
of  the  second  and  third  33 1 ;  find  them. 

11.  Find  3  Nos.  whose  common  difference  is  1,  such  that  the  pro- 
duct of  the  second  and  third  exceeds  that  of  the  first  and  second  by  J. 

12.  The  first  term  is  n'-  n  +  1,  the  common  difference  2  ;  find 
the  sum  of  n  terms. 

13.  How  many  strokes  a-day  do  the  clocks  of  Venice  make, 
which  strike  from  one  to  twenty -four  ? 

14.  How  many  strokes  does  a  common  clock  make  in  12  hdVirs  ? 
and  how  many,  if  it  strikes  also  the  half-hours  ? 

15.  A  debt  can  be  discharged  in  a  year  by  paying  one  shilling 
the  first  week,  three  the  second,  five  the  third,  &c. :  required  the 
last  payment  and  the  amount  of  the  debt. 


118  ARITHMETICAL,    GEOMETRICAL,    AND 

16.  One  hundred  stones  being  placed  on  the  ground  at  the  dis- 
tance of  a  yard  from  one  another,  how  far  will  a  person  travel, 
who  shall  bring  them,  one  by  one,  to  a  basket,  placed  at  the  dis- 
tance of  a  yard  from  the  first  stone  ? 


144.  Quantities  are  said  to  be  in  Geometrical  Pro- 
gression^ when  they  proceed  by  a  common  y^^^^r. 

Thus  1,  3,  9,  &c.  4,  1,  i,  &c.  -^,  *5,  -\^,  &c.  «,  ar,  ar\  &c.  are 
in  G.  p.,  the  common  factors  or  ratios  (as  they  are  called)  being 
3,  J,  - 1,  r,  respectively,  which  may  be  found  by  dividing  any 
term  hy  the  term  preceding, 

145.  Given  a  the  first  term  and  r  tlie  common  ratio 
of  a  GEOM.  series^  to  find'  1  the  n^^  terin  and  S  the  sum 
of  n  ter7ns. 

Here  the  series  will  be  a^  ar^  af^  ar"^^  &c.,  where  the 
index  of  r  in  any  term  is  just  less  hj  one  than  the 
number  of  the  term :  thus,  in  the  2"^^  term  we  have  7*, 
i.  e.  r\  in  the  S""*^,  r'',  in  the  4^^,  /•',  &c.,  and  so  in 
the  n^^  term  we  shall  have  r^'^;  hence  I  =  ar"~\ 

Again  S  =  a    +  ar  +  a?'^  +  &c.  +  ar^'\ 
and  .-.  rS  =  ar  +  a?'^  +  a?'^  -f  <fec.  +  a?*^ ; 

.'.  rS-/S=  ar'^-a,  the  other  terms  disappearing  ; 

T  cy      ar''^-a        r^-\  rl-a     .  , 

hence  o  = =  a ,  or  = ,  snice  rl=a?''^, 

r-1  r-1  r-1 

Ex.  1.  Find  the  6*'*  term  and  the*  sum  of  G  terms  of  1,  2,  4,  &c. 

Here        «=1,  r=2,  ri  =  G; 

.-.  Z  =  1  X  2«-^  =  1  X  2^^  =  1  X  32  =  32;  and  S=  -i^  =  63. 

Ex.  2.  Find  the  S'^  term  and  the  sum  of  8  terms  of  81,  -27,  9,  &c. 
Here  a  =  SI.  r  =  -  J,  7i  =  8  ; 

.•.Z=81x(-i)-3^x-l=_l=-l;and5=  ^-j^-GO|^ 

Ex!  3.  Find  the  sum  of  3  -  C  +  12  -  &c.  to  G  terms. 

Here  a  =  3,  r=-2,  7i=G;  therefore,  without  finding  Z, 


HAKMONICAL   PROGRESSION. 


119 


Ex.  4.  Find  the  sum  of  1 
Here        a  -  1, 


\     2/  "3^    ^    3^ 

.-.  iS'=lx     _4     1    ~_4    i""^?"- 


-  f  +  y^  -  &c.  to  4  terms. 
r=  -f,  71  =  45 


256-81 


175 

'7.3»' 


25 
"27* 


Ex.  5.  Find  the  sum  of  2|  -  1  +  f  -  &c.  to  5  terms. 


3157 
14.5"^ ' 


1201 


Find  the  last  term  and  the  sum  of 


1.  1  +  4+16  +  &C.  to4  terms. 
3.  3+.6+12+&C.  to  6  terms. 
5.  1-4+16-&C.  to  7  terms. 

Find  the  sum  of 

7.  i+^+_i_+&c.  to  8  terms. 
9.  l  +  l+f +&C.  to  6  terms. 
11.  9-6+4-«fec.  to  9  terms. 


2.  5+20  +  80  +  &C.  to  5  terms. 
4.  2-4+8-&C.  to  8  terms. 
6.  l-2+2'-&c.  to  10  terms. 

8.  ^+J+f +&C.  to  G  terms. 
10.  3-i+3^-&c.  to  5  terms. 
12.  100^0+16-&c.to5terms. 


146.  If  r  be  a ^ro^^r  fraction,  that  isjif  r  be  <l,its 
powers,  r%  r\  &c.,  r"*  will,  a  fortiori^  be  also  <  1,  and, 
therefore,  ar'^  will  be  <  <x :  hence,  instead  of  writing 
a 


S 


7"-l 


in   whicli  fraction  both  numerator   and 


denominator  are  negative^  we  may  write,  in  this  case, 
a  -  ar^  _    a        ar'^ 
1-r 


S  = 


1-r        1-r     1-7' 

Now  the  greater  we  take  the  value  of  n  (that  is,  the 
more  terms  we  take  of  the  series),  the  less  will  be  the 
value  of  ar"^ ;  and,  by  taking  n  sufficiently  great,  we 
may  get  ar^  as  small  as  we  please,  only  never  so  small 

I  actually  to  vanish.   If  ar"^  vanished,  we  should  have 


.^B  actu 


120  ARITHMETICAL,    GKOMETRICAL,    AND 

the  Slim  of  the  series  =  - —  ;  but  since,  however  small 
1—r 

may  be  the  value  of  ar^^  the  second  fraction  will  never 

actually  become  2ew,  it  follows  that  the  sum  of  the 

series  will  never  actually  reach  the  above  value,  tliough, 

by  increasing  n,  that  is,  taking  more  terms  of  the  series, 

it  may  be  made  to  approach  it  as  nearly  as  we  please. 

On  this  account is  said  to  be  the  Limit  of  the 

1-r 

sum  of  the  series,  a+ar-\-ar^-\-&:Q.^  or  sometimes  (but 

less  correctly)  the  sum  of  the  series  ad  infinitntn. 

It  is  common  to  denote  the  Limit  of  such  a  sum  by  X 

Ex.  1.  Find  the  Limit  of  the  sum  of  the  series  1  +  ^  +  J  +  &c 

Here  «  =  1,  r  =  \\  .*.  2  =  :j — j  =  -  =  2 ;  i.  ^.  the  more  terms  we 

take  of  this  series,  the  more  nearly  will  their  sum  =  2,  but  will 
never  actually  reach  it. 

Ex.  2.  Sum  2Jj  -  ^  +  Jg^  -  &c.  aH  infinitum. 

Herca  =  2i,r=--J;  .-.  2  =       *       =  ^  =  |=  2^,. 

Ex.  67. 
Find  the  Limit  of  the  sum  of  the  following  series : 
1.  4+2+l  +  «&c.  2.  i+i  +  f+&c.       '         3.  j_Jiy  +  gJj_&c. 

4.  l-.\^l-tc.  5.  l-^+i-&c.  G.  l-|  +  3y_&c. 

7.  j+jy+^+&c.        8.  j  +  f+/^+&c.  0.  2-J  +  ^-&c. 

10.  2-lJ+f-&c.        IL  3J  +  2i+H+&c.         12.  _3i  +  lJ-|  +  &c. 


147.  By  means  of  the  equations  of  g.  p.,  we  may 
solve  many  problems  respecting  series  of  this  kind. 
It  is  not,  however,  generally  easy  to  find  n^  when  the 
other  quantities  are  given, because  this  quantity  occurs 
in  the  form  of  an  index.  The  Student  may  be  able  to 
guess  at  its  value  in  the  simple  instances  we  shall 
here  give  ;  but,  in  other  cases,  it  could  only  be  found 
by  the  aid  of  logarithms. 


IIARMONICAL    PROGRESSION.  121 

Ex.  1.  Find  a  geom.  scries,  whose  1'*  term  is  2  and  7'^  term  ^. 
Here  a-2^  1=^:  ^^=7  ;  .-.  -3'2=2^^  a^^^'""  ^  A?  whence  r  =  ±  ^, 

and  the  series  is  2,  ±  1,  i,  ±  ij  &c. 
Ex.  2.  Given  G  the  second  term  of  a  geo.v.  series  and  54  the 
fourth,  find  the  first  term. 

Here  (S=ai\  54=0r':  .*.— -  =  — ,  or  0=r' ;  hence  r=±  3,  a=  -  =  ±  2. 

Ex.  3.  Insert  3  geom.  means  between  2  and  \0\. 
Here  -y-  is  the  5"*  term  of  a  series,  whose  first  term  is  2 ; 
.•.  -y-=2r^  and  r*=yj ;  whence  r=±f ,  and  the  means  are  ±3, 4|^,±6f. 

Ex.  68. 

1.  How  many  terms  of  the  series  2,  -6,  18,  &c.  must  be  taken 
to  make  -  40  ? 

2.  The  fifth  term  of  a  geom.  series  is  8  times  the  second,  and 
the  third  term  is  12 ;  find  the  series. 

3.  The  fifth  term  of  a  geom.  series  is  4  times  the  third,  and  the 
sum  of  the  first  two  is  -  4 ;  find  the  series. 

4.  The  population  of  a  country  increases  annually  in  g.  p.,  and 
in  4  years  was  raised  from  10000  to  14641  souls ;  by  what  part  of 
itself  was  it  annually  increased  ? 

5.  The  difference  between  the  first  and  second  of  4  numbers  in 
G.  p.  is  12,  and  the  difference  between  the  third  and  fourth  is  300  ; 
find  them. 

6.  Insert  3  g.  m.  between  2  and  32,  and  also  between  ^  and  128. 

7.  Insert  4  g.  m.  between  -  -^  and  3|,  and  also  between  f 
and  -  5y\. 

8.  The  sum  of  an  infinite  geom.  series  is  3,  and  the  sum  of  its 
first  two  terms  is  2|- ;  find  the  series. 

9.  The  sum  of  an  infinite  geom.  series  is  2.  and  the  second  term 
is  -  I ;  find  the  series. 

10.  If  2^3  1,  be  the  first  and  third  terms  of  a  g.  p.,  find  the  sum 
of  the  series  ad  injinitiim. 


148.  Quantities  are  said  to  be  in  Ilarmonical  Pro- 
gression^ when  their  reciprocals  are  in  a.  p. 

Thus,  since  1,  3, '  5,  &c.j  J.  -  },  -  f,  &c.  are  in  a.  p.,  their 
reciprocals  1,  i,  ^.  &c.,  4,  -4,  - 1,  &c..  are  in  h.  p. 

e 


122  ARITHMETICAL,    GEOMETRICAL,    AND 

The  term  Ilarmonical  is  derived  from  the  fact  that  mnsical 
strings  of  equal  thickness  and  tension  will  produce  harmony  when 
sounded  together,  if  their  lengths  be  as  the  reciprocals  of  the  ar. 
eeries  of  natural  numbers,  1,  2,  3,  &c. 

We  cannot  find  the  sum  oi  any  No.  of  terms  of  an 
HARM,  series  ;  but  many  problems  with  respect  to  such 
series  may  be  solved  by  inverting  the  terms,  and  treat- 
ing their  reciprocals  as  in  a.  p. 

Ex.  1.  Continue  to  3  terms  each  way  the  series  2, 3,  G. 

Since  ^j  ^,  ^  are  in  a.  p.  with  common  difference  -  ^, 
the  AR.  series  continued  each  way  is  1, 1,  f ,  ^^  ^,  J,  0,   -  f ,  -  ^  J 
.'.  the  HARM,  series  is  1,  J,  5,  2,  3,  6,  00,  -  6,  -  3. 

Ex.  2.  Insert  4  harm,  means  between  2  and  12. 

We  must  here  insert  4  ar.  means  between  ^  and  y'25  '^hich 
being  ^,  J,  J,  J,  hence  the  harm,  means  required  are  2f ,  3,  4,  6. 
Ex.  69. 

1.  Continue  to  3  terms  each  way,  2,  ^,  1 ;  1^,  2},  3i;  1,  1^,  If. 

2.  Insert  two  h.  means  between  2  and  4,  and  six  between  3  and  ^. 

3.  Find  a  fourth  harm,  proportional  to  G,  8,  12. 

149.  To  find  A,  G,  H  the  ar.,  geom.,  and  harm,  means  between 
ft  and  b. 

(i)  By  (141)  h-A  =  A-a;  ,',  2A  =  a  +  h ,  and  A  =i  (a +  h): 

h       C 
(ii)  by  (144)  tv  =  -  ,  •'.  G^  =  cib^  and   G  =  Va6,  where,  however, 
Cr       a 

unless  a  and  h  have  the  same  sign,  ^ab  will  be  impossible : 

(iii)  by  (148)  \  -  rr  =  4-  --;  '''CtS-.ab=ab-bn,  or  JI  ^  ~ 
o      JJ.      Jj.      a  ^  a+b 

150.  To  prove  that  G  is  the  geom.  m^a;^  between  A  an^  H ;  anc^ 
that  A,  G,  II,  «r6  in  order  of  magnitude^  A  &^m(7  greatest, 

[We  use  the  sign  >  for  greater  than,  and  <  for  less  than.] 

Smce  A  =  -^,  and  J7=  — j,  .\  ^5^=  -^r-  x =  ab  =  G'' i 

2  '  a+b  2         a  +  b  ' 

.'.  C'^  =  ^/AII,  or  6^  is  the  geom.  mean  between  A  and  H, 

41       J       ,-r  •<»  ^  +  ^       2a&         .-    „     ^  ,      •,„      .  , 

Also  A>  U.if  — r-  >  -,  or  if  a^  +  2ab  +  b""  >  4ab, 

2         a  +  b  ' 

or  if  a^  +  b"^  >  2ab ;  and,  this  being  the  case  (137), 

/.  A>  H,  and,  of  course,  >  (r,  whose  value  (being  the  geo.m, 

mean  between  them)  lies  between  those  of  A  and  H. 


HARMONICAL   PROGRESSION.  123 

151.  Three  quantities  a,  b,  c,  are  in  ar.,  geom.,  or  haem.  prog. 

according  as 

a—t     a  a  a 

-. =  - ,  or  =  ^  ,  or  =  - . 

0  -  c     a  0  c 

(\)  4- =   -  =  1  ;  .'.  a—h  =  h  -  c,  and  a.  h,  c,  arc  in  a.  p. : 

^      b  -  C       a  '  ;     ^    ; 

h      c 
(ii)  ah  -h^  =  aJ)  —  ac,  orb^  -  ac ',.'.--  y ,  and  a,  J,  o,  arc  in  o.p.  : 

(iii)  ac-hc-ab—ac^  or,  (dividing  each  by  flSc,)  v = -, 

whence    - ,   -^  ,    -  are  in  a.  p.,  and  therefore  a,  b,  c  are  in  h.  p. 
a^    b       c  ' 

Ex.  70. 

1.  Find  the  ar.,  geom.,  and  harm,  means  between  2  and  4-J. 

2.  Find  the  ar.,  geom.,  and  harm,  means  between  3J  and  IJ. 

3.  The  sum  and  difference  of  the  ar.  and  geom.  means  between 
two  numbers  are  9  and  1  respectively ;  find  them. 

4.  The  HARM,  mean  between  two  numbers  is  |f  of  the  ar.,  and 
one  of  the  numbers  is  4 ;  find  the  other. 

5.  The  dilFerence  of  the  ar.  and  harm,  means  between  two  num- 
bers is  1|;  find  the  numbers,  one  being  four  times  the  other. 

6.  Find  two  numbers  whose  diflference  is  8,  and  the  harm. 
mean  between  them  1|, 


CHAPTER    XI. 

KATIO,    PROPORTION,    AND   VARIATIOIT. 

152.  The  liatio  of  one  quantity  to  another  is  that 
relation  which  the  former  bears  to  the  latter  in  respect 
of  magnitude,  when  the  comparison  is  made  by  con- 
sidering, not  hy  Jiow  much  the  one  is  greater  or  less 
than  the  other,  hut  what  number  of  times  it  contains  it, 
or  is  contained  in  it,  i,  e.  what  multiple^ ;pm%  ovparts^  or, 
in  other  words,  ^NlidXfr action  i\\Q  first  is  of  the  second. 

This  is,  in  fact,  the  way  in  which  we  naturally,  and,  as  it  were, 
unconsciously,  compare  the  magnitude  of  quantities.  Thus  the 
mere  numerical  difference  between  999  and  1000  is  the  same  as 
between  1  and  2 ;  but  no  one  would  hesitate  to  say  that  999 
is  much  greater^  compared  with  1000,  than  1  is,  compared  with  2. 
The  reason  is,  that  the  mind  considers  intuitively  that  999  is  i\ 
much  gi-eater  fraction  of  1000  than  1  is  of  2 ;  and  this  is  what  we 
should  express  by  saying  that  the  ratio  of  999  to  1000  is  greater 
than  that  of  1  to  2.  On  the  other  hand,  we  should  say  at  once 
that  1001  is  much  less^  compared  with  1000,  than  2  is,  compared 
with  1,  the  fraction  in  the  former  case  being  less  than  in  the 
latter. 

The  ratio,  then,  of  one  quantity  to  another  is  repre- 
sented by  the  fraction  obtained  by  dividing  the  for- 
mer by  the  latter. 

Thus,  the  ratio  of  6  to  3  is  5  or  2,  that  of  15  to  40  is  ||  or  J, 

that  of  4a  to  6Z)  is  ttt  or  -kt 
Co        oh 

Of  course  the  two  quantities  compared  (if  they  are 
not  mere  numbers,  or  algebraical  quantities  express- 
ing numbers)  must  be  of  the  same  kind,  or  one  could 
not  be  a  fraction  of  the  other. 


RATIO,    PROPORTION,    AND    VARIATION.  125 

Thus,  the  ratio  of  £9  to  £12  is  the  same  as  that  of  9  cwt.  to 
12  cwt.,  or  of  9  to  12,  or  of  3  to  4,  or  of  |  to  1 ;  since,  in  each  of 
these  pairs  of  quantities,  the  first  is  J  of  the  second,  and  hence 
J  is  the  value  of  each  of  these  ratios ;  in  saying  which  we  may- 
suppose,  if  we  please,  a  tacit  reference  to  1,  i.  e.  in  saying  that  the 
ratio  of  £9  to  £12  is  f ,  we  may  either  imply  that  £9  is  -J  of  £12, 
or  that  the  ratio  of  £9  to  £12  is  the  same  as  that  of  f  to  1. 

153.  Tlie  ratio  of  one  quantity  to  another  is  ex- 
pressed by  two  points  placed  between  tliem,  as  <^  :  J ; 
and  tlie  former  is  called  the  antecedent  term  of  the 
ratio,  the  latter  the  conseqiient, 

A  ratio  is  said  to  be  a  ratio  of  greater  or  less  in- 
equality, according  as  the  antecedent  is  greater  or  less 
than  the  consequent. 

The  ratio  of  a^ :  JMs  called  the  dujplicate{i,e,  squared) 
ratio  of  ct  :  h,  a^  :  1/  the  triplicate  ratio  of  a  :  5,  &c. 

151:.  Problems  upon  ratios  are  solved  by  represent- 
ing them  by  their  corresponding  fractions,  which  may 
now  be  treated  by  the  ordinary  rules. 

Thus  ratios  are  comjyared  with  one  another,  by  re- 
ducing the  corresponding  fractions  to  common  den", 
and  comparing  the  num" ;  and,  if  these  fractions  be 
multiplied  together,  the  resulting  fraction  is  said  to  bo 
the  ratio  com/po^indedofthe  ratios  represented  by  them, 

Ex.  1.  Compare  the  ratios  5  :  7  and  4  :  9. 

■^^^^-  Ifj  11 ;  whence  5  :  7  >  4  :  9. 

Ex.  2.  Find  the  ratio  of  4  :  f.  Ans.  -?  -*-  J  =  -f  x  J  =  ||. 

Ex.  3.  What  is  the  ratio  compounded  of  2  :  3,  G  :  7,  14  :  15  ? 

Ans.  f  X  4  X  II  =  yy  or  8  :  15. 

155.  A  ratio  of  greater  inequality  is  diminished, 
and  of  less  inequality  increased,  by  adding  the  same 
quantity  to  both  its  terms. 

For  -  ^  - — ^-.  as  ah\ax  ^ cib-rhx.  as  ax    hx.  as  dJ  ^  5. 
h  <  h-{-x  <  '  <     '  < 


126  RATIO,   PROPORTION,   AND   VARIATION. 

In  like  manner  it  may  be  shewn  that  a  ratio  of 
greater  inequalit}^  is  increased,  and  of  less  diminished, 
by  subtracting  the  same  quantity  from  both  its  terms. 

Ex.  71. 

1.  Compare  the  ratios  3:4  and  4  :  5  ;  13  :  14  and  23  :  24 ;  3:7, 
7  :  11,  and  11 :  15. 

2.  Ofa+i  :  a-l)  and  a^  +  l"^ :  a^-V^^  which  is  >,  supposing  a  >  hi 

3.  Which  is  loss  of  x  +  y  :y  and  4x:x  +  yl  ofx^+y^-.x+y  and 
x^+y^  :x'^  +  y'^7  oix^  +  y'^  and  x'^  +  y^ :  x^-x^y+x^y^-xy^+y*7 

4.  Find  the  ratio  compounded  of  3:5,  10 :  21.  and  14 :  15 ; 
of  7  :  0,  102 :  105,  and  15  :  17. 

e    -n-  J  XT.       X-  11/.    a'^  +  ax+x'^  ,  cr-cix+x^ 

5.  Fmd  the  ratio  compounded  of  ^ — — r  and . 

a'^-a^x+ax-x^  a+x  ^ 

G.  Compound  x""-  9a;  +  20  :  x''-(jx  and  a;''-13a;  +  42  :  x''-  bx. 

7.  Compound  the  ratios  a+l)  :  a-b^  a'  +  h^ :  (a+hy,  (a^-hy  :  a*-h\ 

8.  What  is  the  ratio  compounded  of  the  duplicate  ratio  of 
a+h-^a  -h,  and  the  difference  of  the  duplicate  ratios  of  fl^ :  a 
and  a  :  &,  supposing  a>h1 

9.  What  quantity  must  be  added  to  each  term  of  the  ratio  a  :  J, 
that  it  may  be  equal  to  the  ratio  c:dl 

10.  Shew  that  a-h-.a  +  1)      a*  -  5' :  a^  +  6^,  according  as  a  :  &  is 

a  ratio  of  less  or  greater  inequality. 

15G.  When  two  ratios  are  equal^  the  four  quantities 
composing  them  are  said  to  be  proportio7ial  to  one 

another:  thus,  iia  :  h=c :  d,  i,e.  if  ^  —  -  ,  then  a,  5,  c?,  c?, 

are  proportionals.  This  is  expressed  by  saying  that 
a  is  to  h  as  c  is  to  d^  and  denoted  thus,  a  :  i  : :  c  :  d. 

The  first  and  last  quantities  in  a  proportion  are 
called  tlie  Extremes^  the  other  two  the  Means, 

Problems  on  proportions,  like  those  on  ratios,  are 
solved  by  the  use  of  fractions. 

157.  When  four  quantities  are  projyortionals^  the  pro- 
duct of  the  extremes  is  equal  to  the  product  of  the  means. 

For  if  _  =  -^ ,  then  ad  =  le. 
0       d 


RATIO,    PROPORTION,    AND   VARIATION.  127 

Hence,  if  three  terms  of  a  proportion  are  given,  we 
can  find  the  other  ;  thus 

1)G     7       ad  ad     -,      Ig 

d  G  h  a 

Cor.  li a:h::l):  g^  then  ac  =  V, 

158.  Jf  the  product  of  two  q^iantities  he  equal  to  that 
of  two  others^  the  four  are  proportionals^  those  of  one 
product  ieing  the  extremes^  and  of  the  other  the  means. 

For  if  ad  =  he.  then  -v  =  -^?  or  -  =  -  ; 
0      d       G      d 

and   .\a\l)\\G\d^  or  a\  g::1):  d^  in  which   propor- 
tions <2,  d  are  the  extremes,  and  5,  g  the  means. 
^0  \i  ac  =^  V^  a  :!) :  :!)  :  c. 

159.  If  3  quantities  are  prop^%  the  first  has  to  the 
third  the  duplicate  ratio  of  that  which  it  has  to  the  secon  d. 

■r.      .p^      5  ^1        a      a      h      a      a      c^ 
i^  or  It  ^  =  -,  then  -  ^  ~  y^ -■=.--  y^  ~  —  -~\ 

0         G  G         0        G        0         0        0 

.\  a\c  is  the  duplicate  ratio  of  a  :  5  (153). 

160.  When  four  magnitudes  are  proportionals^  if  any 
equijmdtiples  whatever  he  taken  of  tJiefrst  andthird^ 
and  any  whatever  of  the  second  and  fourth^  then^  if  the 
midtiple  of  the  first  5^  >,  =r,  <  that  of  the  second^  the 
multiple  of  the  third  shall  he  >,  =,  <  that  of  the  fourth. 

y^      ...a      G         ,         ma      mc     ,  . 

hov  it  -  =  ~,  we  have  — j-  =  —-.where  manan  may 
0      d  no       nd 

be  any  quantities  whatever ;  and  hence  it  follows  that, 

if  m^  >,  =,  <  n5,  so  also  is  mc  >,  =,  <  nd, 

161.  Conversely,  If  there  he  four  magnitudes  such^ 
that^  when  any  equimultiples  whatever  of  the  first  and 
third  are  talcen,  and  any  lohatever  of  the  second  and 
fo^irih^  it  isfound^  that  if  the  multiple  of  the  first  he 
>j  r=j  <  that  of  the  second^  that  of  the  third  is  always 


128  RATIO,    PEOPORTION,    AND   VARIATION. 

>5  =5  <  that  of  the  fourth^  then  these  four  quantities 
are  projyortionals. 

For,  let  «,  5,  c,  d  be  such  that,  any  equimultiples, 
ma^  mc^  being  taken  of  tlie  first  and  third,  and  any 
nl)^  nd^  of  the  second  and  fourtl),  it  is  found  that  ac- 
cording as  7na>^  =,  <nb^  so  also  is  mc>^  =,  <Qid\ 
and  let  e  be  the  fourth  proportional  to  a,  5,  c, 

mi  »        a      G        Qua      mc  r.       n       ,  ^ 

Inen,  since  ^  =  ->  •*•  — r  =  —  ^^^  f^H  values  of  m 
b      e        nb        ne 

and  n ;  suppose  m  and  n  to  be  taken  such  that  ma=^nl)^ 

then  also  772(?  =  -yif? :  but  when  ma  =  nb^  by  our  hyp., 

a      c 
mc  =:  nd ;  hence  72cZ  =  ne.  or  cZ  =  ^ ;  and  /.  ^"=15  <^^' 

6      a. 

<a:>,  5,  c,  cZ  are  proportionals. 

162.  If^ :  b  : :  c :  d,  and  b  :  e  : :  d :  f,  then  a :  e : :  c :  f. 

■r^     a      G       ^b     d,       abed       a      c 

For  -  =  -,  and  -  =:-. ;  .-.  ^  x  -  =  -  x  -.,  or  -  =  2- 

b      d  e    f        b     e      d     f        e     f 

This  is  the  proposition  ex  ceqtiali,  referred  to  in  Euc.  v. 

163. 7)^a :  b:  :c :  d,  a^id  e :  f :  :g :  h,  then  ae :  bf : :  eg :  dh. 

^     a      c        1  e      q        ae      eg 

This  is  called  coinpounding  the  two  proportions,  and 
so  we  may  compound  any  number  of  such  proportions. 

164.  If  4  quantities  form  a  proportion,  we  may 
derive  from  them  many  other  proportions,  all  equally 
true. 

Thus,  if  ^  =  - ,  then  —^  =  -^,  or  772a  :mb::c:d: 
b      d  mb      d 

similarly 

nia  :b\:  mc :  c7,  a\  mb  :  :  <? :  md^  a:b: :  mc:  md ; 

J  •    VI  a     b  J        b  d     g 

and,  in  like  manner,      \  ~  \  \  c:  a.  a :  ~ : :  c :  —.   cxrc. ; 

711  on  m  m 

that  is,  either  the  first  orfoicrth  terms  of  any  proportion 


RATIO,    PROPORTION,  AND  VARIATION.  120 

may  be  multiplied  oz-  divided  by  any  quantity,  pro- 
vided that  either  the  second  or  third  be  multiplied  or 
divided  by  the  same. 

Hence  we  may  get  rid  of  fractions,  when  occurring  in  propor- 
tions, by  multiplying  the  1'*  and  2"*^,  or  1'*  and  3'**,  &c.  terms  by 
the  L.  c.  M.  of  their  den";  thus,  if  -Ja :  yV^-o"- Aj  (multiplying 
!•»  and  2"^  by  36,  3'^  and  4"*  by  200),  we  have  4a :  36 ::  15  :  16. 

165.  Again,  all  the  results  of  (85-88)  may  be  ap- 
plied to  proportional  quantities. 

Thus,   if  a:h::c:dj  then  inv.,  h:  and:  c^  or  alt^   a:c::h:  d* 
so  also,  a  ±d:  a::c  ±  d:  c^  a  ±h:h::C  ±  d:d^ 
a  ^b  :  a  -  d :: c+d :  c  -  dj  ma  +  7i5  :  ma  —  nb - mc  +  nd-.  mc  -  nd^  &c. 
with  similar  prop",  having  a",  &",  c",  ^",  in  the  place  of  a,  &,  c,  d. 

In  hke  manner,  if  a-.hv.c-.d-e :/:: &c.,  by  which  it  is  meant 
that    a:h  ::  cid^    or    a-.b  ::  e:/,    or    c:d  ::  e :/,    &c.,    so    that 

c-  =  -^  =  -  =  &c.,  then  we  have  a:  h-.-.a  +  c  +  e+kc. :  l  +  d  +/+&c. ; 

that  is,  If  any  quantities  he  in  continued  proportion^  as  one  of  the 
antecedents  is  to  its  consequent,  so  is  the  sum  of  all  the  antecedents 
to  the  sum  of  all  the  consequents. 

So  also  a-.h  ::  ma  +  nc  -^  pe   +  &c. :  mb  +  nd    +  pf  +  &c., 
a"  :  &'*::7/^a**  +  nc"  -f-^e"  +  &c. :  ml**  +  nd""  +  pf"  +  &c., 
with  other  similar  proportions,  which  may  be  proved  as  in  (88). 

Ex.  1.  Find  a  fourth  proportional  to  ^,  J,  and  J. 

Since  ^  =  -,  (157)  this  is  ^-^  =  4. 
a  i 

Ex.  2.  Find  a  mean  proportional  to  2,  and  8. 

Since  6^  =  ac,  (157)  this  is  V(2  x  8)  =  ^16  =  4, 

Ex.  3.  If  a  :  b=c  :  d,  express  (a-¥d)-(b+c)  in  terms  of  a,  h,  c  only, 

/       hcX    ;,     .     a^-ab-ac+bc      (a-b)  (a-c) 
Here  (a+d)-(b  +  c)=[a-\-  -  \-(b+c)= = -^^ — 


Ex.  72. 

1.  Find  a  fourth  proportional  to  3,  5,  6 ;  to  12,  5,  10 ;  to  f,  £,  |. 

2.  Find  a  third  proportional  to  4.  6  ;  to  2,  3  ;  to  f,  f . 

3.  Find  a  mean  proportional  to  4,  9 ;  to  4,  || ;  to  IJ,  1  j^. 

4.  Ua:b::h  :  C,  tllCU  a*  +  5'  :  a  -^  C ::  tt^  -  2»*  :  a  ^  C. 

6* 


130  RATIO,    PROPORTION,    AND   VARIATION. 

5.  If  I  =  ~  shew  that  (a  +  I)  (c  +  d)  =^j(c  +  dy^^(a  +  h)\ 

G.    If  «  :  Z> ::  C  :  cl,  and  171  :n::p  '•  q, 

then  ma  +  nh  :  ^wa  —  nZ»  -pc  +  qd  -.pc  —  qd, 

7.  If  a  :  5  ::  Z)  :  C,  thcil  a^  -  h^  :  a::h^  -  c"^  :  C. 

8.  If  a  :  5  ::  C  :  iZ ::  <?  :/,  then  Ct  -  6 '.  J)  -/::  C  :  d, 

9.  If  a  :  & ::  &  :  c,  then  ma^  -  nh^ :  ma  -  nc  ::pa^  +  ql^  -.pa  +  ^'c. 
10.  If  r^:  &::  ?»:  ^,  then  a-  25  +  c  =  ■^''~  ^^'  -  ^^  "  ''^' 


a  c 


11.  If  ?  =  ^,,  then  [^   ^^-\-  A^  .  1\  -  0^_*)  0»_rj) 


6      (i'  \a      dj       \h      ej  abc 

12.  If  a  :  Z>=&  :  c,  then  a  +  Z>  +  c  :  «  -  Z>  +  c ::  Ta  +  &  +  c)^ :  a*  +  Z<^  +  <J^ 

13.  Solve  the  equations 

(i)  ^x  ■¥  ^1)1  ^x  -^h  -'a:!).     (\\)  x  -^^  a'2x-'b':Zx  +  h -.Ax-  a, 
(iii)  iB  +  2/  +  l:a;.+  i/  +  2::G:7  ) 

y  ^2x'.y  -2x'.:l2x  +  C>y  -Z:(jy-\2x~\\' 
(iv)  a;:27::j^:9::2:a;-2/. 

14.  What  number  is  that  to  whicli  if  1,  5,  and  13  be  severally 
added,  the  first  sum  shall  be  to  the  second  as  the  second  to 
the  third  ? 

15.  Find  two  numbers  in  the  ratio  of  2-J- :  2,  such  that,  when  di- 
minished each  by  5,  they  shall  bo  in  that  of  1 J  :  I. 

IC.  A  railway  passenger  observes  that  a  train  passes  liim,  mov- 
ing in'  the  opposite  direction,  in  2",  whereas,  if  it  had  been 
moving  in  the  same  direction  with  him,  it  would  have  passed 
him  in  30"  :  compare  the  rates  of  the  two  trains. 

17.  A  and  B  trade  with  different  sums  :  A  gains  £200,  B  loses 
£50,  and  now  yi's  stock  :  i>'s ::  2  :  ^  ;  but,  if  A  had  gained 
£100  and  B  lost  £85,  their  stocks  would  have  been  as  15  :  3^  ; 
find  the  original  stock  of  each. 

18  A  hare  is  50  leaps  before  a  greyhound,  and  takes  four  leaps 
to  his  three ;  but  two  of  the  greyhound's  leaps  are  as  much 
as  three  of  the  hare's  :  how  many  leaps  must  the  greyhound 
take  to  catch  the  hare  ? 

19.  Divide  £500  among  A^  B,  G  in  the  proportion  of  3,  4,  5,  and 
also  in  the  proportion  of  J,  ^,  | ;  and  if  ^4's  portion  be  to 
J5's::9  :8,  and  to  (7's::G:  5,  shew  that  the  shares  of  J[,  B,  0 
are  in  the  proportion  of  H.  IJ,  1|. 


RATIO,   PROPORTION,    AND   VARIATION.  131 

20.  A  quantity  of  milk  is  increased  by  watering  in  the  ratio  of 
4 :  5,  and  then  three  gallons  are  sold ;  the  rest,  being  mixed 
with  three  quarts  of  water,  is  increased  in  the  ratio  of  6  :  7  ; 
how  many  gallons  of  milk  were  there  at  first  ? 


16S.  The  value  of  any  Alg.  quantity  will,  of  course, 
depend  on  the  values  we  give  to  the  letters  it  contains. 

Def.  "When  two  quantities  are  such,  that  their  ra- 
tio is  constant^  that  is,  remains  the  same,  wliatever 
values  we  give  to  the  letters  they  contain,  one  of 
them  is  said  to  vary  as  the  other. 

The  sign  used  to  denote  variation  is  Gc(read  varies  as). 

Thus,  x^  +  3a;  a  ^x""  +  (Sx^  since  -■- — —  =-.  - ,  whatever  be  the 
value  of  X. 

167.  Hence  if  -4  oc  ^,  (where  A  and  B  are  used  to 

denote,  not  numerical  or  constant^  but  algebraical  or 

variable  quantities,  such  as  admit  of  diflferent  values  by 

giving  different  values  to  the  letters  they  contain)  then, 

according  to  the  above  definition,  the  value  of  the  ratio 

A  :  B  will  remain  constant,  whatever  may  be  the  values 

of  the  quantities  A  and  B  themselves.  If  then  weputm 

A 
todenote  this  constant  value,  we  have—  =m,  or  A=mB ; 

SO  that,  whe7i  one  quantity  varies  as  anotlier^  they  are 
connected  hy  a  constant  Qnvltiplier, 

Thus  ic'+  3a;  =  ^(2aj^  +  Ga;),  from  which  it  follows  necessarily  that 

.VT-TT-  =  vi  •  for  ali  values  of  x,  or,  as  above  stated,  x^  +  Zx^z  2x'^  +  6x, 
2aj*  +  6a;     2'  ^     '  ' 

168.  Hence  also  if  J.  gc  ^,  and  a,  5,  be  any  pair  of 
values  of  A  and  B,  tlien  for  any  other  values  of  ^  and  B, 
we  have  A  :  B  =  77i  =  a  :  h,  that  is,  when  mie  qiicmtity 
varies  as  another^  if<^^y  two  pairs  of  values  he  taken  of 
them^  the  four  willhe  2yroportionals :  pr  since  A\a\:B:h^ 
we  may  state  tins  by  saying  that  if  one  of  them  be 


182  RATIO,    PROPORTION,    AND   VARIATION. 

changed  from  any  one  value  {A)  to  any  other  value  («), 
the  other  will  be  changed  in  the  same  proportion  {vom 
the  value  (B)  corresponding  to  the  first  to  the  value  (J) 
corresponding  to  the  second. 

169.  The  following  are  terms  used  in  Variation  : 

1.  If  A=mB^  then  A  is  said  to  vary  directly  as  B; 

2.  If  J.  =  ~,  A  is  said  to  vary  inversely  as  B; 

3.  If  ^  =:mBC,  then  A  is  said  to  vary  jointly  as  J?  and  (7; 

4.  l{A=7n  --- ,  then  A  is  said  to  vary  directly  as  B^ 
and  inversely  as  C, 

170.  ThefollowingresultsinVariationarenoticeable. 
(i)  If  ^  oc  ^  and  B  a:  C,  then  Ack  C 

For  letal=m^,  B=nC\  then  A=rnnC\  and  .•.  ^  oc  {?, 
since,  77i,  ?i,  being  constant,  so  also  is  mn. 

So  also,  if  J.  a  J?  and  B  qc~^  then  J.  oc  — ^ . 

(ii)  If^oc  CandBcK  C\A±Bcc  C,and^{AB)cx:  C. 

For  let  J.  =  7?i{7,  B=:nC\ 
then  Adc:B~mC±nC={m±7iyC,  and  .\^1±j5oc  C 
and  V  (^^)  =  V  (m6'  x  nC)  =  V  (m7i(7^)  =  V  (m/i)  6; 
and  therefore  V  (^^)  c/:  (7. 

(iii)  If  Act,  BC\  thenBa:-,  and  6^oc  4- 

For  Iet^=mi?6^,  then  B=  1.:^,  or  ^oc  4.;  so  (7a:  4 

(iv)  If  ^  cr.  ^,  and  Co:  D,  then  .4(7oc  BD, 
For  let^=mj5, 6^=?ii>;  then7l6'=w?ji?i>,  or^C^cc  BD. 
(v)  If^oci?,  then  JL^»oc^». 
(vi)  If  J.  cc  jB,  and  7^  be  any  other  quantity, 

then  ^7^  X  ^7^,  and  4^-^r 


RATIO,   PROPORTION,    AND   VARIATION.  133 

171.  -5^  A,  B,  C,  he  variable  quantities^  depending 
on  07ie  another^  and  it  is  observed  that^  when  C  is  Ttept 
constant^  KccH^  and  lohen  B  is  hejpt  constant^  A  cc  0  ; 
tken^  generally^  that  is,  when  all  three  are  allowed  to 
change  their  values  together,  A  (X  BO. 

For  since  Ao:  B,  when  C  is  kept  constant,  A  must 
be  of  the  form  mB,  where  m  is  some  constant,  and 
7nay,  therefore,  contain  the  constant  C\  but  not  B. 

[From  this  we  see  that  A  must  contain  ^  as  a 
factor,  but  not  B"^,  B^,  &c.,  and  may  contain  C] 

Again,  since  A  cc  (7,.when  B  is  kept  constant,  A 
must  be  also  of  the  form  nC,  where  n  is  some  coii- 
slant,  and  ma.y,  therefore,  contain  the  constants?,  but 
not  C. 

[From  this  we  see  that  A  must  contain  C,  as  a 
factor,  but  not  C,  C%  &c.,  and  7nay  contain  B,  as,  in 
fact,  w^e  have  ah^eady  shewn  it  does,] 

Upon  the  whole^  then,  it  appears  that  A  must  con- 
tain both  B  and  C  as  factors,  but  no  other  powers  of 
B  or  C,  and  therefore  must  be  of  the  form  ^^^6', 
where  ^  is  a  constant,  containing  neither  ^  nor  C; 
hence,  since  A  =pBC,  we  have  A  qc  BC,  when  all 
three  are  allowed  to  change  their  values  together. 

The  above  result  may  similarly  be  proved  for  any 
number  of  quantities,  B,  C,  D,  &c. ;  so  that,  if  any 
quantity  vary  separately  as  each  ofseveral  others,  when 
tiie  rest  are  kept  constant,  it  varies  as  their  product, 
when  all  are  allowed  to  change  their  values  together. 

Ex.  1.  If  a  oc  y^c.  and  1,  2,  3,  be  contemporaneous  Tallies  of 
a,  &j  c,  express  a  in  terms  of  &  and  c. 

Since  a  oc  V^c,  .'.a-  mVc,  where  we  have  to  find  m ;  now,  when 
^=2  and  c=3.  a  becomes  1;  .*.  l  =  12w.  or  w^yj,  and  .*.  a^^¥c. 


134  RATIO,  PROPORTION,  AND  VARIATIO'N. 

Ex.  2.  If  y  =  the  sum  of  two  quantities,  one  of  which  oc  x  and 
the  other  ,oc  x^^  and  when  ic  =  1,  y  =  6,  when  a;  =  2,  y  =  20 ;  express 
y  in  terms  of  x. 

Here  y  =  mx  +  ??x',  where  we  have  to  find  m  and  /i . 
noW;  by  the  Question,  when  a;  ^  1,  y  =  6,  .*.  (i)  6  =  «»i  +  n, 
and  when  ic  =  2,  y  =  20,  .*.  (ii)  20  =  2/?!  +  4/1 ; 
from  which  equations  w  =  2,  n  =  4,  and  .•.  y  =  2a;  +  4a;'. 

Ex.  73. 

1.  If  xy  cc  X'  +  2/')  ^^d  ^j  4,  be  contemporaneous  values  of  x 
and  y^  express  xy  in  terms  of  x"^  +  y-, 

2.  If  y=  the  sum  of  two  quantities,  whereof  one  is  constant  and 
the  other  oc  x  inversely^  and  when  a;  =  2,  ?/  =  0,  when  a;  =  3,  y  =  1, 
find  the  value  of  y^  when  x  =  (j. 

3.  If  2/  =  the  sum  of  two  quantities,  whereof  one  is  constant, 
and  the  other  xy^  and  when  a;=2,  y  =  -  2J.  when  a;  =  -  2.  i/=l, 
express  y  in  terms  of  x. 

4.  If  y  =  the  sum  of  three  quantities,  which  vary  as  a;,  a;',  «* 
respectively,  and  when  aj  =  1,  2,  3,  y  =  6,  22,  54  respectively,  ex- 
press y  in  terms  of  x, 

5.  If  2/  =  the  sum  of  three  quantities,  of  which  the  first  oc  a;*^, 
the  second  oc  a;,  and  the  third  is  constant;  and  when  a;  =  I,  2,  3, 
y  -  G,  11,  18,  respectively,  express  y  in  terms  of  a;. 

6.  Given  that  zee  x  -^  y^  and  y  cc  a;-,  and  that  when  x-\^  the 
values  of  y  and  s  are  *  and  J,  express  z  in  terms  of  a;. 

z  y  11 

7.  If  a;  cc  -  and  z"^  oc  ~,  shew  that  a;  <x  -  oc  -. 

y  ^  y     z 

8.  The  area  of  any  triangle  varies  jointly  as  any  side,  and  the 
perpendicular  let  fall  upon  it  from  the  opposite  angle ;  express 
the  area  of  the  right-angled  triangle  ABGm  terms  of  the  sides 
AG^  BG^  containing  the  right  angle,  it  being  found  that,  when  the 
sum  of  the  two  sides  is  14  feet  and  the  hypothenuse  10  feet,  the 
area  is  24  square  feet. 


CHAPTEK  XII. 


172.  The  Variations  of  any  No.  of  quantities  are 
the  different  arrangements  wliicli  can  be  made  of 
them,  taking  a  certain  No.  at  a  time  together. 

Thus  the  Var""  of  a.  5,  c^  two  together,  are  db^  ha^  ac,  ca,  hc^  ch. 

"When  all  are  taken  together,  tlie  Var"*  are  called 
Permiitati(nis :  but  this  distinction  is  not  always  ob- 
served, the  words  Variation  and  Permutation  being 
used  by  some  as  synonymous. 

173.  The  No,  of  Var""'  of  n  different  things^  taken  r 
together^  is  n  (n  - 1)  (n  -  2) (n  -r  + 1). 

Let  there  be  7i  different  things,  a^  5,  <?,  d^  &c. 

The  No.  of  Var"'  which  can  be  formed  of  these  n 
things,  taken  singly^  is,  of  course,  n. 

Now  let  us  remove  a\  there  will  then  be  n-\  things, 
J,  (?,  d^  &c.,  and  the  Var"'  of  these  taken  singly^  will  (as 
before)  be  n  -  1.  If  then  we  set  a  before  each  of  these, 
there  will  be  n-\  Var"^  of  ?i  things,  a^l^e^d^  &c.  taken 
two  and  two  together,-  in  which  a  stands  first;  similarly 
there  will  be  n-1  such  Var"',  in  which  I  stands  first ; 
and  so  of  the  rest :  therefore,  on  the  whole,  there  will 
be  n{n-l)  Var"'  of  n  things  taken  two  and  two  together. 

Let  us  again  remove  a ;  there  will  be  n  -  1  things, 
J,  Gy  d^  &c.,  and  the  Var"'  of  these,  taken  two  and  two 
together,  wdll  be  {n-1)  {n-2)  by  what  precedes  ;  and, 
by  the  same  course  of  reasoning,  it  will  appear  that, 
on  the  whole,  there  will  be  n  {n-1)  {n-2)  Var"'  of  n 
things  taken  thi^ee  and  three  together. 


136  VARIATIONS,   PERMUTATIONS, 

Suppose  then  this  law  to  hold  for  the  No.  of  Var™ 
of  n  things  a^  5,  c^  d^  &c.  taken  r  - 1  together,  which 
would  be,  therefore,  n  {n-1)  {n-2) ....  \n- {r-l)-{-l\^ 
or  n {n -1)  {7i-2) . . .  .{n-r  +  2). 

Now  remove  a ;  there  will  then  be  n-1  things  J,  c,  d, 
&c.,  and  the  Yar"'  of  these,  taken  r-1  together,  would 
be  found  from  the  preceding  result,  by  writing  in  it 
n-1  for  n,  and  would,  therefore,  be 

{n-1)  {n-2) (n-r  +  1). 

If  now  we  set  a  before  each  of  these,  there  would  be 

(n-1) (?i-2) (n-r  +  1)  Yar"^  of  n  things  a, 5, c,  d, 

&c.  taken  r  together,  in  which  a  stands  first }  similarly, 
"when  h  stands  first,  and  so  of  the  rest :  therefore,  on  the 
whole,  there  would  be  n {n-1)  {n- 2) . . .  .  {n-7'  + 1) 
Yar"^  of  n  things  taken  r  together. 

If  then  the  formula  represent  correctly  the  No.  of 
Yar"^  of  n  things  when  taken  r-1  together,  it  would 
also  when  they  are  taken  r  together ;  but  we  have 
shown  it  to  be  true  when  they  are  taken  1,  2,  or  3 
together  ;  therefore  when  taken  4  together ;  and, 
therefore,  when  5  together,  &c.,  that  is,  it  is  generally 
true  for  all  values  we  can  give  to  r, 

174.  Hence  denoting  by  F„  F„  V,,  &c.  F,  the  No. 
of  Yar"'  of  n  things  taken  1,  2,  3,  &c.  r  together,  we 
have,  from  the  preceding  formula, 

F,  =  ?2,    V,=:n{n-1\    F3  =  n(;i-)l(7i-2),  &c. 
Vr=n{n-1) {n-r  +  1). 

Cor.  If  r=?^,  or  all  the  quantities  are  taken  together, 
then  the  No.  of  Perm"^(P)  of  n  things,  is 

n{n-l)  (?i-2) . . .  {n-n+1)  =  n  {n-1)  (72-2)  .  . .  1 ; 
or,  reversing  the  order  of  the  factors, 
P=  1.2.3 n. 


AND   COMBINATIONS.  137 

175.  The  No.  of  Penn"^  of  w  letters^  whereof  ^  am 
aV,  q  are  b'^,  r  are  c'c?,  (&c.  is 

1.2.3  . . .  .  n 

1.2.3  ....  p  X  1.2.3 q  X  dtc! 

For  let  iV^be  the  No.  of  such  Perin""\  Suppose  now 
that  in  any  one  of  them  we  change  the  j9  ah  into  differ' 
ent  letters ;  then  these  letters  might  be  arranged  (174:* 
Cor.)  in  1.2.3. . .  ,p  different  ways,  and  so  instead  of 
this  one  Perm",  in  which  ^  letters  WQuld  have  been  a's, 
we  shall  now  have  1.2.3  ...^:>  different  Perm"^  Tlie 
same  would  be  true  for  each  of  the  iV^Perm"' ;  hence, 
if  tlie^  a'§  were  changed  to  differentlQitQr^^  we  should 

have  altogether  1.2.3 p  x  iV  different  Perm"^  of  7i 

letters,  whereof  still  q  are  5's,  r  are  c's,  &c. 

So  if  in  these  the  q  Vs  were  changed  to  different 

letters,  we  should  have  1.2.3 qx  1.2.3 . ,  ,  ,2?xN 

different  Perm"^  of  ti  things,  whereof  still  r  would  be  o's, 
and  so  we  may  go  on  until  all  the  n  letters  are  differ- 
ent ;  but  when  this  is  the  case  we  know  (174.  Cor.) 

that  their  whole  number  of  permutations=1.2.3 n ; 

hence  1.2.3 . . .  .^  x  1.2.3. . . .  ^  x  &c.  x  iV^=1.2.3 ....  n, 

.  -j^_ 1.2.3.... 71 

anc  i\  -  ^^^  _  _^  X  1.2.3  ....(/  x  i^c' 
^  Ex.  1.  How  many  changes  can  be  rung  with  5  bells  out  of  8  ? 
How  many  with  the  whole  peal  ? 

Here  V,  -  8.  7.  6.  5.  4  =  6720,  P  =  8.  7.  G.  5.  4.  3.  2. 1  =  40320. 

Ex.  2.  How  many  different  words  may  be  made  with  all  the 
letters  of  the  expression  a^l'c  ? 

1  2-3  4  5  6 

Of  these  6  letters,  3  aro  a'^,  and  2  ¥s  j  .-.  IT=  i~9  3 Vl  ^^  "  ^^' 

Ex.  3.  What  No.  of  things  is  that,  whereof  the  No.  of  Yar"*, 
taken  3  together,  is  20  times  as  great  as  the  No.  of  Yar""  cf  half 
the  same  No.  of  things  taken  2  together? 
,    Here,  if  n  denote  the  No.  of  things  required,  we  have 
n  (71  - 1)  (71  -  2)  =  20  {\n)  (i?i  -r  1),  whence  n  =  6. 


138  VARIATIONS,    PERMUTATIONS, 

Ex.  74. 

1.  How  many  changes  may  be  rung  with  5  bells  out  of  G,  and 
how  many  with  the  whole  peal? 

2.  In  how  many  diflferent  ways  may  7  persons  seat  themselves 
at  table  ? 

3.  How  many  different  words  may  be  made  of  all  the  letters  of 
the  words  division^  insincere^  commencement,  haccalaureus  ? 

4.  How  many  different  words  may  be  made  of  the  letters  of  the 
expression  a^h^c'dl 

5.  The  No.  of  Yai*-,  3  together  :  the  No.,  4  together  : :  1 :  6  ;  find 
the  No.  of  things. 

G.  How  many  diffei'ent  words  may  be  made  of  all  the  letters  of 
the  words  mammalia^  carCtvansera^  Oroonolo^  Mississippi  ? 

7.  The  No.  of  things :  the  No.  of  Var",  3  together : :  1  :  20 ;  find 
the  No.  of  things. 

8.  The  No.  of  Yar'"  of  n  things,  3  together :  the  No.  of  Var"  of 
71  +  2  things,  3  together : :  5  :  12 ;  find  n, 

9.  The  No.  of  Var*"  of  n  things,  4  together :  the  No.  of  Var""  of 
f  71  things,  4  together  : :  13  :  2 ;  find  n» 

10.  If  the  No.  of  Var**'  of  n  things,  3  together,  be  12  times  as 
great  as  the  No.  of  Yar^  of  \n  things,  3  together,  what  is  the  No. 
of  Perm"^  of  the  same  n  things  ? 

11.  Of  what  No.  of  things  are  the  Perm^'  720  ? 

12.  There  are  7  letters,  of  which  a  certain  No.  area's  ;  and  210 
different  words  can  be  made  of  them ;  how  many  a's  are  there  ? 


176.  The  Corahinations  of  any  No.  of  quantities  are 
the  different  sets  that  can  be  made  of  them,  taking  a 
certain  No.  together,  without  regard  to  the  order  in 
which  they  are  placed. 

Thus,  the  Comb*"  of  c^,  &,  c,  (Z,  3  together,  are  dbc^  abd,  acd,  led. 

It  is  readily  seen  that  each  Coiiib^  will  supply  as 
many  corresponding  Far"',  as  the  No.  of  quantities  it 
contains  admits  of  Perm"'. 

Thus,  the  ComV  ale  supplies  the  1.2.3  or  G  Var"  dhc^  acl,  laCj 
hca,  cab,  cla. 


AND   CO^IBINATIONS.  139 

177.  The  No,  of  ComTf^  ofn  different  things^  taken  r 
together^  is 

n(n-lUn-2)  •_!_•  -  (n-r  +  1) 
i.:4.5  .  .  7  .  r 
For  (176)  each  Coriib^  of  r  things  will  supply  1.2.3  „.r 
Var'^'  of  r  things;  hence,  if  C^  denote    the   No.  of 
Comb"''  oin  things,  v  together,  we  have 

1.2.3 ry^Cr  —  No.  of  Var"'  of  n  things,  r  together 

z:^  Y^:=n{ii-\)  (7^-2)....(n.--r  +  l); 

•*•  '  r:2.3  ....?• 

rx       TT          n     't'^  n      n{n-Y)  ^    n{n-l){7i-2)  J. 
CoR.  Hence  C  =  -,  C,=  --^^\  ^*"~^2  3 ' 

Now  it  will  be  seen  hereafter  that  these  are  the 
same  as  the  coefficients  of  the  binomial  (1+a?)",  so  that 

(1  +  j^)^  =  1  +  c,x  +  cy+  &c.  +  c\x\ 

Hence,  putting^=l, we  have2'*=l+C;+C;+&c.+C;; 
or  the  sum  of  all  the  Comb"'  that  can  be  made  of 
n  things,  taken  1,  2,  3,  &c.  n  together  =  2^-1. 

178.  The  expression  for  (7;.,  (by  multiplying  both  num'^ 
and  den*^  by  1.2.3  ... (?i-r))  may  be  put  into  the  form 

n{7i-l){n-^) (71-r+l)  X  (yi-r) 3.2.1 

1.2.3 r  X  1.2.3 {n-r) 

1.2.3 n Y- 

~  1.2.3  ....  7"  X  1.2.3  ....  {n-7')~'  |^r  \7i~r 

if  we  use  \n  to  denote  the  continued  product  1.2.3  . . . -^i. 
Hence,  writing  n-  r  for  ?',  Ave  have 

^nr  — =z ^r  J 

\9l-r   \7'        (7^1^1-r 

or  the  No.  of  Comb""  of  n  things  taken  n-7'  together 
=  the  No.  of  them  taken  r  together. 

The  Comb"^  of  one  of  these  sets  are  said  to  be 
Buppleineniary  to  those  of  the  other. 


140      VARIATIONS,  PERMUTATIONS,  AND  COMBINATIONS. 

Ex.  1.  Find  the  No.  of  Comb"  of  10  things,  3  and  G  together  ? 
„      ■  10.9.8     ,^^       ,  .,      ^      10.9.8.7     ^.„ 

Here  d  =   ^tt^  "  ^^^'  ^'^^^  ^"  =  6'4  =  yr^-g-^  ^  210. 

Ex.  2.  How  many  words  of  G  letters  might  be  made  out  of  the 
lirst  10  letters  of  the  alphabf^t,  with  two  vowels  in  each  word  ? 

In  these  10  letters,  there  are  7  consonants  and  3  vowels ;  and  in 
each  of  the  required  words,  there  are  to  be  4  consonants  and  2 
vowels :  now  the  7  consonants  can  be  combined  four  together  in 
35  ways,  and  the  3  vowels,  two  together,  in  3  ways  ;  hence  there 
can  be  formed  35  x  3=  105  different  sets  of  G  letters,  of  which  4 
are  consonants  and  2  vowels :  but  each  of  these  sets  of  G  letters 
may  ha 2^ermuted  6.5.4.3.2.1  =  720  ways,  each  of  these  forming  a 
different  ^cord^  though  the  whole  720  are  composed  of  the  same  G 
letters :  hence  the  No.  required  =  105  x  720  =  75600. 

Ex.  75. 

1.  How  many  Comb'"' can  be  made  of  9  things,  4  together?  how 
many,  G  together  ?  how  many,  7  together  ? 

2.  How  many  ComV'  can  be  made  of  11  things,  4  together? 
how  man}^,  7  together  ?  how  many,  10  together  ? 

3.  A  person  having  15  friends,  on  how  many  days  might  he 
invite  a  different  party  of  10  ?  or  of  12  ? 

4.  How  often  might  a  common  die  be  thrown,  so  as  to  expose 
five  different  faces  ? 

5.  Find  the  whole  No.  of  Comb"  of  G  things,  1,  2,  &c.,  6  together. 
G.  Four  persons  are  chosen  by  lot  out  of  10 ;  in  how  many  ways 

can  this  be  done  ?  and  how  often  would  any  one  person  be  chosen  1 

7.  How  often  may  a  different  guard  be  posted  of  G  men  out  of  GO '] 
on  how  many  of  these  occasions  would  any  given  man  be  taken  / 

8.  The  No.  of  Comb"  of  \n  things,  2  together,  is  15  ;  fmd  n. 

9.  The  No.  of  Comb"  of  n  things,  3  together,  is  y\  of  the  No., 
5  together  ;  find  n. 

10.  The  No.  of  Comb"  of  t?  +  1  things,  4  together,  is  9  times  the 
No.  of  Comb"  of  n  things,  2  together ; .  find  n. 

11.  The  No.  of  Comb"  of  hi  things.  4  together,  is  3J  of  the  No. 
of  Comb"*  of  ^n  things,  3  together  ;  find  7i. 

12.  How  many  words  of  6  letters  may  be  made  out  of  the  26 
letters  of  the  alphabet,  with  2  out  of  the  5  vowels  in  every  word  ? 


CHAPTER   XIII. 


THE   BINOMIAL   THEOREM. 


179.  The  Binomial  Theorem  is  a  formula,  discovered 
by  Sir  Isaac  Newton^  by  means  of  wliicli  any  binomial 
may  be  raised  to  any  given  power,  without  going 
through  the  ordinary  process  of  Involution.  It  may 
be  stated  as  follows  :  AVhatever  be  the  value  of  n^ 
positive  or  negative,  fractional  or  integral, 

j-^ a    .^  +  Otc. , 

where  the  coefficient  of  <x""^a;^= — ^ -hlLi L 1 

1.2  ...  r  ' 

and  this,  being  the  coefficient  of  the  {r+lj-^  term  of  the 

expansion,  where  r  may  represent  a?2?/ positive  integer 

whatever,  is  called  the  coefficient  oit\\Q  general  term. 

It  will  be  noticed  that  the  coefF^  of  x^  x^^  &c.  x^^  in 
the  above,^w^lien  n  is  a  jjositive  integer^  are  no  other 
than  the  ISTos.  of  Combinations  of  n  things  taken 
1,  2,  etc.  7',  together.  On  this  account  we  will  use 
the  letters  Cj,  C2,  &c.  C^^  to  denote  these  coefP  in  all 
cases  ;  and  so  we  may  write  the  formula 

(^+a;f =^«+  C,a''-^x-\-  (7,^"  V+  &c.+ 0^-^a?^+&c. 

In  this  expression,  a  and  ccmay  stand  for  any  quan- 
tities whatever ;  so  that 

Or-xY—  \a-^{-x)Y:=.a''^r  C,cO'-^  {-x)-\- C^a''-''  (-ir)'+  &c. 
^a''-C,a''-^x      +C;a^-V      -&c., 
where  the  terms  ^Y^alternaiehj^^o^\\AN^  and  negative: 
and  (1  ± xY^ \±G,x^  Csc'±C,x^  +  C,x'  ±  &c. 


^ 


142  THE   BINOMIAL  THEOREM. 

180.  To  prove  the  Binomial  Theorem  when  the  index 
is  a  rosiTivE  integek. 

"We  shall  find,  by  actual  multiplication,  that 
{x  +  a)  (x  -^T))  ^x^  -^  {a  +  1)  X  +  db^ 
{x  +  a)  {x  -^  V)  {x  ^-  c)  =  a;'  +  (a  +  Z>  +  C)  x^  +  {ah  +  ac  +  he)  x  -r  abc. 
Assume  this  Law  of  Formation  to  hold  for  ?i  -  1  factors,  so  that 
(x  +  «i)  (x  +  ai)...{x  +  ti„i)  =  a;**"^+jpia;""^  +  p^x"" "  +  &c.  +^n-i) 
where  ^i=«i+^a+«3  +  &c.j  iH=aiai+aiaz-^a-iaz  +  &c..  &c.  =  &c. 

2>n\=  aia^az ...  a„  i  , 
then,  multiplying  by  another  factor,  x  +  (X„,  we  have 

(;c+ai)  (aj+a2)...(a;+fl^„)=a;''+^iic""^+    jp2a;"^  +  &c.+  i)^^x 

+  ct^a;**'*  +^ia„a;"'''  +  &c.  +7?„  ^a^x  -^PmCtn 
=a;"+2'xa;"'*+    $'2ic"^  +  &c.+    $'n.ia;+       5-, 
where  qi=^pi  +  «»=  «i  +  «a  +  «3  +  &c.  +.^„, 

q,2.  -pi  +  ^iCt„  =  ai^a  +  <^^i«3  +  <?2«3  +  &c.  +  ai<7„  +  a3^„  +  &c. 
&c.  =  &c. 

that  is,  if  the  Law  holds  for  the  product  of  ?2.  -  1  factors,  it  holds 
also  for  that  of  n  factors :  but  we  have  seen  above  that  it  does  hold 
for  three  factors,  therefore  for  foui\  and  therefore  iov  five^  and  so 
on  ;  that  is,  it  holds  generally,  when  n  is  a  positive  integer. 

Now,  it  is  easily  seen  that  the  terms  in  <7i,  $'2,  q^^  &c.,  are  the 
different  Comb''"  of  the  n  letters  a^  «2,  ^3,  &c.  «„,  taken  one^  two, 
three,  &c.  together;  and,  consequently,  the  No.  of  terms  in  qi  is  61, 
in  qa  is  Ca,  &c.,  as  in  (177).  Let  us  put  a  for  each  of  rti,  «2,  &c. : 
then  the  first  side  becomes  (x  +  a)",  and  each  of  the  terms  in 
$'1,  ^'a,  ^3,  &c.  becomes  cr,  a^,  a',  &c.  respectively ;  and  therefore 
we  have 

(x  +  ay  =  ic"  +  Cicix'''  +  C^a^'x'' '  +  &c. 

n        ,    n(n  —  1)    „     .      „ 
=  X''  +  :,-  ax""'^  +  -^Y^ — -  «'a:"*'  +  &c. 

And,  of  course,  it  will  follow  in  like  manner,  that 
(a  +  a*)**  =  a"  +  Cxxa''  ^  +  C2ic-a***  +  &c. 


THE   BINOMIAL   THEOREM.  14:3 

181.  There  are  only  n-\-\  terms  in  the  expansion  of 
(1  +xY^  when  the  index  is  a  positive  integer. 

Since  the  coeff.  of  the  (r+l)*^  term=  (7;.,  we  see  that 
if  r  be  such  that  the  last  factor  of  the  num'',  7i-r+l =0, 
then  the  (r-j-l)"^  and  all  the  following  terms  (all  of 
which  would  involve  this  factor)  will  vanish,  i,  e. 
the  series  will  have  ended  with  the  r^^  t3:m.  Now  if 
n-r  +  1  =0,  then  r  ==  ?i  +  1 ;  and  iv,<)  series  will 
have  ended  with  the  (^2/  +  1)^^  term. 

182.  In  the  expansion  of{l+xy^  the  coeff ^  of  terms, 
equally  distant  from  the  beginning  and  end,  are  the 
sAMEj  when  the  index  is  a  positive  integer. 

The  (r  + 1)^^  term  from  the  end  (having  r  after  it) 
will  be  the  \{n-^V)-rY^  or  (^i-r-^Xf"  from  the  begin- 
ning, and  its  coeff.  will  therefore  be  (7„_^;  but,  (178) 

n{n-\)„,{n^^V)__      [^     _     [^     _ 
1.^...?  \r\n-r     \n—r\r 

or  coeif.  of  (t'+I)^^  term  from  beginning  =  coeff.  of 
(r+1)^^  term  from  the  end. 

N.  B.  The  uumber  of  terms,  7^+1,  being  odd  or  ex>en  as  n  is  e^eii 
or  odd^  it  follows  that,  if  n  be  even,  there  will  be  one  middle  term, 
but  if  odd,  two  middle  terms,  which,  by  (182),  will  have  equal 
coeff',  and  on  each  side  of  which  the  same  coeff*  will  occur  in  order. 
When,  therefore,  in  expanding  a  binomial  with  a  'positive  integral 
index,  we  have  passed  the  middle  term  or  terms,  we  shall  find  all 
the  coeff*  repeating  themselves  ;  and,  instead  of  calculating  those 
of  the  remaining  terms,  we  may  write  down,  in  inverted  order,  the 
coeff"  already  found,  as  in  the  following  examples. 

Ex.  1.  (1  +if)*  =  l  + ja;  +  |^ic^-{-&c.  =  1  +  4a;  +  Go;^  +  4^' +  a;*. 

We  shall  not,  however,  give  any  more  examples  of  the  3"*,  4* 
and  5*^  powers  of  a  binomial,  which  the  Student  should  be  able  to 
write  down  as  in  (42). 


144  THE   BINOMIAL   THEOREM 

-^     ^    .1      ^T     1      '^       7.G    ,     7.6.5    3     ^ 

Ex.  2.  (i-xy  =  1  -  i  ^+  J72  ^  -  1:2:3 ^  ■"  ^^• 

=  l-7aj + 2lx^-Z5x^  +  35a;^-21aj* + 7a;«-aj^ 

=729x«-G  X  243aj'^  x  ly  + 15  x  81x*  x  i2/'-20  x  27x»  x  {y* 

■i-15x9x^x^*-6xZxx^y^+-^y^ 
=729x'-729x'y-i-^-^^xY-H-^Y+^^Y 

Ex.  76. 

1.  (1  +  xy.        2.  (a  +  ic)^          3.  (1  -  xy.  4.  (a  -  xy. 

5.  (1  +  xy\       6.  (1  -  2xy\       7.  («  -  3ic)«.  8.  (2aj+a)». 

9.  (2c^-3a;)'.    10.  (l-ixy\      11.  (1-iic)^^  12.  aa;-^y)^^ 


183.  To  2^^'ove  the  Binomial  Theorem^  lohen  the 
index  is  fractional  or  negative. 

It  will  be  sufBcient  if  we  can  prove  the  Theorem 
for  the  expansion  of  (1+ ct')"^,  that  is,  if  we  can  shew 
that  iov  all  values  of  n,  (l+a:^y'=  l+C^x-\-C^d'-\-  &e. 


For  then,  since  0^  +  x  =  a\l+  -]  ,  we  shall  have 

—  a^+  C^a^'^x-r  C,a"V  +  (fee,  as  required. 

Let  then  the  series  1+  —a?+     '\    — ^a?''+&c.,  what- 
1  1.2 

ever  be  the  value  of  7?i,  be  denoted  by  the  symbol^/* (771). 

Now,  when  ra  is  a  positive  integer,  we  know  that  this 

series  represents  the  expansion  of  (1  +  x)^^  that  is, 

/(m)  =  (l+ajy%  when  in  is  a  positive  integer.    We 

shall  now  shew  that  this  is  the  case  for  all  values  of  m. 


THE   BINOMIAL   THEOREM.  145 

mce  f  {rri)—\'\--x-\ \—- — -  x^  +  &c. 

1  1.2 

and  /(m)  x/(;0=l+p'«+  '^-^-  x'  +  &c. 


+ 

'^      1 

mn 

a?' 

+  &C. 

1 

n  {n  — 
1.2 

1) 

a;' 

+  &C. 

VI 


l-i-ccx+        h^  +  &c. 

where  we  use  a,  5,  &c.  to  denote  the  coeff%  found  by 
addition,  of  a?,  a?',  tfec,  so  that 

-f  n^  I  =  -1-2  "^  ^^^^  "^  12' 
Now  5,  0,  &c.  might  be  reduced  to  much  simpler 
forms  than  these,  but  the  process  would  be  tedious :  we 
may  find  them  however,  immediately,  by  the  following 
consideration.  Since  the  above  multiplication  does  not 
at  all  depend  upon  the  actual  values  of  m  and  n,  we 
should  still  have,  by  the  addition,  the  same  values  as 
above  for  a^  5,  &c.,  whether  77i  and  tI' stand  for  positive 
or  negative,  integral  or  fractional,  quantities. 

But  when  m  and  n  2iXQ positive  integers^  we  know  that 
/(m)  =  (l+irr,  f{n)  =  {l+xr, 
and  .\f{m)  x /(»)=(!  +a;)'»  x  (l+a;)"=(l +«)'"■'"; 
and  since  m  +  n  is  here  a  positive  integer,  we  know 
also  that 
(l+a.r-=l  +  "TLJ^x^  {m  +  nUm+n-D  ^,  ^  ^^_ 

1  l.J 

Here,  therefore,  we  have  the  values  of  ^,  5,  &c.  when 
m  and  n  are  positive  integers :  hence  also  they  will 
7 


146  THE    BINOMIAL   THEOEEM. 

be  the  same,  whatever  be  the  values  of  m  and  n,  and 
we  have,  therefore,  in  all  cases, 

or,  since  this  series  would  be  denoted  by/  (m+n),  we 
have/* (771)  x/*(7i)=y(m+7i),  for  all  values  of  m  and  n. 

The  student  may  easil}''  satisfj^  himself  that  the  values  just 
obtained  for  a,  Z>,  c,  &c.  are  identical  with  the  former,  though 
simplified  in  form ;  thus 

m  (m  - 1)  n  in-X)      m  (m  -  1)  +  2m  n  +  n  (n  -  1) 

h  =  -~j-  ^nm^  "j;2~  " TS 

_m(m-l  +  n)  +  7i  {m^n-\)  __  {m  +  n)  (m  -i-  n  - 1) 

Hence  /(m)  x  /(?i)  x  f{p)=^  f{m  +  n)  x  f(p)  = 
f{m-{'7i  +i>)?  and  similarly  for  any  No.  of  such  fac- 
tors ;  i.  e,  the  product  of  any  two,  or  more,  such  series, 
as  that  denoted  byy(m),  produces  another 'series  of 
precisely  the  same  form, 

Now,  (i),  let  there  be  n  factors,  each=/[—V  where 

m  and  n  are  positive  integers;  then 

since  m  is  a  positive  integer; 

,  in  "hW 

/.  taking  the  n^^^  root  on  both  sides,  (!+«?)«  =/(— 

Hence/ (m)  is  the  series  for  (1  +  x)''\  so  long  as  the 
index  \^  positive^  whether  it  be  integral  ox  fractional. 

Again,  (ii),  let  n^=^-7n^  where  in  \^  positive^  but 
may  be  integral  or  fractional;  then 

/  W  x/  (-  m)  =/  {vi  -  m)  =/(0)  =  1, 


THE    BINOMIAL   THEOEEM.  147 

(since  the  series  becomes  =  1,  if  we  put  0  for  m  in  it); 

1  1  .  .  . 

,-.  fi-iii)  =  — , — r  =  Tz r-  i  Since  m  is  positive, 

fip)   (i+^-r 

z=z  il+x)'"^^  bj  the  Theory  of  Indices. 
Hencey(-7?^)  is  the  series  for  {\+xy"\  where  the  in- 
dex is  neffative,(iiid  maybe  QitliQv  integral  or  fractional. 
It  follows  then  that  for  all  values  of  the  index,  we  have 

{l^xf  =f{n)  =  1  +  %  +  ^'l^izl)  of  +  &c. 

J.  1.  J 

184.  We  have  seen  (181)  that,  Avhen  the  index  is  a 
positive  integer,  this  series  will  stop  after  n+1  terms; 
when  fractional  or  negative,  it  will  never  terminate,  but 
consist  of  an  infinite  number  of  terms,  since  we  cannot 
then  find  any  value  of  r,  which  Vvill  make  7i-r-\-l=0. 

Ex.  1. 


(l+oj)  ^^=1+ -—-a;  +  — ^ 


2  (-2-1)   ,     -2  (-2-1)  (-2-2) 


1.2.3 


&c. 


2       2.3        2.3.4 
=1-  IT  x+  Yo^^~  tVq  ^^  "^  ^^'  =  I  -  2x  +  2x'  -  ix^  +  &c. 

In  this  Ex.  there  is  some  trouble  in  simplifying  coefT',  and 
getting  rid  of  superfluous  signs :  to  save  this,  it  will  be  useful  to 
remember  the  result  of  the  following  general  example. 

Ex.  2. 

a.xr-i-^x.:^^^t^  &c. 

Ex.  3.  (l-^x)-'  =  l-~x  +  jl  x'-  ~~^  x'  +  &c. 
=  l-3x  +  6x''-  10j;=  +  &c. 

Ex.  4.  (1 .  ..)i=  1 .  f ...  iiiri> .' .  ^^^  ^  &c. 


148 


THE  BINOMIAL  THEOREM. 


Here  also  it  will  be  well  to  notice  the  following  general  results, 
Ex.  5. 

(1  ±  a;)7  =  1  ±   j^a;  +  — j;^—  x''  ^  I.2.3  * 

So  also 
(\±x)  ?=l±^-a;+  -T4r4  «  ^  — ^".r^— 3-^ ic'  +  &c (m). 


Ex.  G.  (1^^^  =  1.30..  j-^:^ 


2.5.8 


1.2.3.3« 
1  +  fa;  +  |aj^  +  f  ja;''  +  &c. 


iC*  +  &c. 


Ex.  77. 

1.  (l+ar)-\        2.  (l-3a;)-\        3.  (l  +  3ic)-'.       4.    (l-2icr». 

5.  (l-ia;)-«.      6.  (l  +  Ja;)-».        7.  (l  +  2;7j)t        8.  (1  -  3jr)* 

9.  (l-xyk      10.  (1  -  a;^)^.      11.       ^      .        12. 

Vl  -  a; 


vr 


Ex, 
Ex.8 


:.,.<..^,-...-.(.4)-.,t-|(5).g(iy-.o.( 


-  8a-^a;  +  40a-''aj''  -  KjOar'x^  +  ifcc. 


,.-.,i..^(.-|)-L.-iju|(5).a(-?)-..c.i 

-f^.      .cc     15  aj^      35a;»       . 
(  a      2  a'      2  a' 

=a*2  +  Sa"2^  +  -ya"2ic-  +  -^-a'^v^  +  &c. 

Ex.  78. 

1.  i2-x)-\  2.  (3-2a;)-*.     3.  (a+&a')-».      4.  (a-5»a!)-«. 

6.  (a"^-J^-«.       C.  (a'-x^)^,    7.  (a'K&"V'.  8-  (a-x)K 
9.  (a*-x')'y.        10.  (a'-a;')^.  11.  (a'-aj»)"*     12.  (aa;-2«)"*. 


I 


CHAPTER    XIY. 

NOTATION,    DECEVIALS,    INTEREST,    &C, 

185.  Notation  is  the  method  of  expressing  numbers 
by  means  of  a  series  of  powers  of  some  one  fixed  num- 
ber, which  is  said  to  be  the  radix  or  hase  of  the  scale^ 
in  which  the  different  numbers  arc  expressed. 

Thus  in  common  Arithmetic,  all  Nos.  are  expressed  in  a  scale 
whose  base  is  10 ;  for  3578  denotes  3000  +  500  +  70  +  8,  i.  e, 
3.10^  +  5.10^  +  7.10  +  8 ;  so  also  370,  when  expressed  in  a  scale 
\vhos3  radix  is  12,  is  274,  since  2.12^^  +  7.12+4=288 +  84+4-376. 

186.  Ifx  he  any  integer^  any  No,  "^may  he  expressed 

in  the  form  ]Sr=:p"r"+pn.ir"^+&c.+p2r+PiT+Po5'^^^^^'^ 
the  coefficients  p^,  Pn.i,  &c.  are  integers  all  less  than  r. 

For  divide  iV"  by  the  greatest  power  of  r  it  contains, 
suppose  r**;  and  letthe  quotient  be/>,j  (which  will,  of 
course,be  <r),and  the  remainder  N^ :  i\\Q\\N=^j>  r'^+iV^p 

Similarly  N,=^]y,,.{^-'''''+N,,N,=:p,,.,r'''''VN,,&^^^  and 
thus  continuing  the  i>rocess  until  the  rem""  becomes  <r, 
j?^  suppose,  we  have  iT^^^r"*  +^Vi  ^'"'"-^  +  ^^'  +i>3>''* 

Some  of  the  coefficients  j^o,  ^„  j^i'i  ^^'  ^^"^^7  vanish, 
but  none  can  be  >  r.  Their  values  then  may  range 
from  0  to  r-1,  and  these  different  values  are  called 
the  digits  of  the  corresponding  scale.  Ilence^  includ- 
ing zero^  there  will  be  r  digits  in  the  scale  of/*. 

Thus  in  the  scale  of  12,  the  digits  will  be  0,  1,  2,  3,  4,  5,  C,  7,  8, 0, 
i  and<?,  where  t  and  e  are  used  to  denote  the  digits  10  and  11. 

187.  In  the  Binary  scale,  tlie  radix  is  2  ;  in  the 
Tarnary^  3  ;  in  the  Quaternary'^  4  ;  in  the  Qxdiiaryy  5  ; 
in  the  Senary^  0,  &c. ;  in  the  Denary  ov  Decimal^  10; 

7* 


150  NOTATION,    DECIMALS,    INTEREST,    &C. 

in  the  Undenary^  11 ;  in  the  Duodenary  or  Duodeci- 
mal,  12 ;  &c. 

All  Nos.  are  supposed  to  be  expressed  in  the  com- 
mon or  denary  scale,  unless  the  contrary  is  mentioned. 

188.  To  express  any  proposed  No.  in  a  given  scale. 

Let  iV^be  the  given  No.  which  is  to  be  expressed  in 
thescale  otV,  in  the  form  N=p,{i''^+&Q.+p^r''+p{r'+p^ : 
Ave  are  to  shew  how  the  digitsj?>„,  p^.^^  &c.  may  be  found. 

Divide  iTby  r  ;  then  we  shall  have 

j^^p^r-'  +  &(t.+p,r+p,+^, 

L  e.  we  shall  have  an  integral  quotient,  ^„r'***4-&c.+pj 
(=:iV^jj  suppose,)  withremainder^j)^;  hence  the  remain- 
der, upon  dividing  iT  by  ?',  is  j^^,  the  last  of  the  digits. 
Again,  divide  iV^i  by  r ;  then  we  shall  have 

V  r  r 

hence  the  rem^,  upon  dividing  N^  by  r  is^^i,  the  last  hut 
one  of  the  digits ;  and  so  dividing  N^  by  r,  we  get  jpg,&c. 
Ex.  Express  the  common  number  3700  in  the  quinary^  and 
convert  37704  from  the  nonary  to  the  octenary  scale. 

Ex.2.     8)37704 


Ex.  1.     5)  3700 

5)  740  ...  0 

5)  148 ...  0 

5)  29 ...  3 

5)  5.. .4 

Ans.  104300.  1...0 


8)  4311...  5 

8)  480 ...  1 

8)  54  ...4 

6...1 


Ans,  C1415. 


Notice  that  in  Ex.  2,  the  radix  is  9,  and  therefore,  when,  in 
beginning  the  division,  we  arc  obliged  to  take  the  two  figures  37, 
these  do  not  mean  tldrty-seccn^  but  Zx9  +  7=thirty'fonr :  hence 
8  in  37  will  go  4  times  with  2  oyer  3  3  in  27  (not  ticenty-seve/i, 
but  2^^ ^1  ^ticenty-Jlve)  will  go  3  times  with  1  overj  and  so  on. 


NOTATION,   DECIMALS,   INTEREST,    &C, 


151 


Ex.  79. 

1.  Express  1828,  34705  in  the  septenary  scale. 

2.  Express  300  in  the  scales  of  2,  3,  4,  5,  6. 

3.  Express  10000  in  the  scales  of  7,  8,  9,  11,  12. 

4.  Transform  444  and  4321  from  the  quinary  to  the  septenary. 

5.  Transform  27^  and  7007  from  the  undenary  to  the  octenary. 

6.  Transform  123  and  10000  from  the  nonary  to  the  quaternary. 


189.  The  common  processes  of  Arithmetic  are  car- 
ried on  with  these,  as  with  ordinary  Nos.,  observing 
that  when  we  have  to  find  what  K^os.  we  are  to  carry 
in  Addition,  (fee,  we  must  not  now  divide  by  10,  bnt 
by  the  radix  of  the  scale  in  question. 


Ex.1. 


Addition, 

r=:4 

r  =  7 

32123 

65432 

21003 

54321 

33012 

43210 

22033 

1444 

31102 

65001 

332011 


226041 


201210 
102221 

21212 


Suhtrnction. 
3_  r=12 

7^8 
5^6^4 


1^864 


Ex.  2.  Multiply  together  68  and  71  in  the  undenary  scale ; 
express  also  and  multiply  these  Nos.  in  the  nonary  scale,  and 
compare  the  results,  by  reducing  each  to  the  other  scale. 
Here  68  and  71  in  the  undenary  =  82  and  86  in  the  nonary : 
68  82  9)  4378  11)  7823 

71  86  9)633...3  11)  642.. .8 

f  543  9)  65...2  11)  52...7 

1?L  1^  "7...8  X.S 

4378  7823 

It  will  be  seen  that  in  the  last  two  operations  we  have  shewn 
that  4378  in  the  undenary  =  7823  in  the  nonary,  and  mce  versdy 
as  it  should  be. 

Ex.  3.  Divide  234431  by  414  (quinary),  and  extract  the  square 
root  of  122112  (senary). 


414)  234431 
2302 

(310 

122112(252 
4 

423 
•  414 

45)  421 
401 

'       41 

542)  2012 

1524 

44 

There  is  a  rem'  here  in  each  case. 


152  NOTATION,    DECIMALS,    INTEREST,    &0. 

Ex.  80. 

1.  Take  six  terms  of  the  series  1,  10,  10^,  &c. ;  express  and  add 
them  in  the  senary  scale,  and  reduce  the  result  to  the  denary. 

2.  ]Multiply  the  common  Nos,  G4  and  33  in  the  binary  and 
quaternary,  and  transform  each  result  to  the  other  scale. 

3.  Transform  175G  and  345  from  the  octenary  scale  to  the 
nonary ;  multiply  them  in  both  scales,  and  divide  the  result  in 
each  case  by  the  first  of  the  two  numbers. 

4.  Divide  51117344  by  G75  (octenary),  37542027  by  42t  (ud- 
denary),  and  29^96580  by  2tt9  (duodenary), 

5.  Extract  the  square  roots  of  25400544  (senary),  47610370 
(nonary),  and  32^75721  (duodenaiy). 

6.  Express  in  common  Nos.  the  greatest  and  least  that  can  be 
'  formed  with  four  fiprures  in  the  scales  of  G,  7,  and  8. 


190.  A  decimal  fraction  may  be  considered  as  a  vul- 
gar fraction,  whose  den"^  is  some  power  of  10,  the  ISiO, 
of  decimal  places  pointed  off  from  the  right  being  the 
same  as  the  index  of  the  den^  Hence,  if  P  represent 
the  digits,  or,  as  they  are  called,  i\\Q  sigyiificant  party 
of  a  decimal  of ^  places,  its  equivalent  vulgar  fraction 

It  is  obvious  that  decimals,  having  the  same  sig- 
nificant part,  P,  may  difi'er  much  in  value,  in  conse- 
quence of  the  difference  in  the  value  of  j(>,  i.  e,  in  the 
position  of  their  decimal  points. 

Thus  1.23  =  J-?f,  -^123  ==  1|?   12.3  =  Ig. 

191.  To  prove  the  rule  for  j^ointmg  in  MuJP  of 
Decimals, 

Let  Jfand  iV^be  two  fractions,  which,  expressed  as 
decimals,  give  the  significant  parts  Pand  (>,  with  jt> 
and  q  places  of  decimals  respectively  ;   then 

M^^~,  N^  J-,  and  Jfxi\^=  —  x  ^^E^, 

l()p^    .       IC?  10^      10^       10^  +  ^' 


KOTATION,    DECIMALS,    mTEREST,    «feC.  153 

PO 

Now  — -^  represents  a  decimal,  whose  significant 

part  is  PQ  (tlie  product  of  the  two  decimals  as  whole 
Nos.)  and  liaving  j9+2'  decimal  places  ;  hence  the  rule: 

Multiply  as  in  v)hole  Nos,  /  and  in  the  product 
point  ojf  as  many  decimal  places  as  there  are  in  tJie 
Midtiplier  and  M%dtiplicand  together, 

192.  To  prove  the  rule  for  pointing  in  Div^  of 
Decimals, 

Let  My  JVy  jP,  Q,Pj  q  be  the  same  as  before; 

'^®^iy"~10^  •  10^~10P^    Q       ^'10^' 


M    P     \             P 

p              > 

'n-  Q-iQ^"'''^-  q'""' 

=  ^.10",as_p  =  <7. 

p 

Now  yr  is  the  quotient  obtained  by  dividing  P  by 

Qy  as  in  whole  Nos. ;  hence  the  rule : 

Pivide  as  in  xvhole  Ifos, ;  then 

(i)  If  the  No»  of  places  in  the  Dividend  exceed  that 
in  the  divisor ^  point  off  in  the  quotient  a  iTo.  of  deci- 
vial  places  equal  to  that  excess  y 

(ii)  If  the  No,  in  the  dividend  he  the  same  as  that 
in  the  divisor  ^  the  q^iotient  will  have  no  decimal  places; 

(iii)  If  the  No,  in  the  dividend  fall  short  of  that  in 
the  divisor^  annex  to  the  quotient  a  No,  of  cyphers 
equal  to  that  defect. 

Notice  that  any  cyphers,  annexed  to  the  dividend  in 
the  process  of  Division,  must  be  reckoned  as  so  many 

decimal  places  :  thus  1  ^  12.5  =  -^—-^  =  .08. 

12.0 

193.  To  prove  the  ride  for  reducing  a  circulati7ig 
decimal  to  a  vulgar  fraction. 


154  NOTATION,    DECIMALS,    INTEREST,    SzC, 

We  need  here  consider  only  the  fractional  part  of  a  circulating 
decimal.  If  there  be  any  figures  hefore  the  decimal  point,  these 
may  be  kept  separate,  and  connected  with  the  vulgar  fraction 
equivalent  to  the  other  part,  so  making  a  mixed  No. 

Let  iV^be  a  circulating  decimal,  in  which  jP  repre- 
sents tlie  figures  not  recurring,  and  Q  the  period  or 
recurring  jDart;  and  let  P  and  Q  contain  p  and  q 
digits  respectively. 

Then  iY:=  ,PQQ  &c.  and  10^  .N  =^  P ^QQQ  ^^c. 
and  10  P^'^.N^I'Q.QQQ  ike. 

.'.{lOP-^^-lO^)  jV=PQ-P, 

10?>-f(?_10P      10^(10^-1) 
Hence  the  rule — (since  10^-1  will  be  expressed  by 
q  nines,  and  10^  is  1  followed  hj  2^  cyphers) — 

Por  the  mime'rator,  set  down  the  decimal  to  the  end 
of  the  first  period,  and  subtract  from  it  the  non-recwr- 
ring  part;  and  for  the  denominator ,  set  down  as  many 
9'^  as  there  are  recurring fig%iresy followed  hy  as  many 
cyphers  as  there  are  Qiaii-o'ccurring  figures. 

194.  Let  ^  be  a  proper  fraction  in  its  loioest  terms. 

Then  if  h  can  be  but  in  the  form  2*^  5  ,  i.  e.  the  pro- 
duct of  any  powers  of  2  and  5,  the  fraction  may  be 
reduced  to  a  terminating  decimal,  in  which  tlie  num- 
ber of  places  will  be  the  greater  of  the  two,  m  and  n. 

l^or  it  m  >  ??,  then  — -^r-  =  ^   ^   ■= , 

which,  expressed  as  a  decimal  (190),  has  m  decimal 

places  :  and  it  m  <  ?? ,  tlien  -——  =  — = , 

which,  expressed  as  a  decimal,  has  71  decimal  places. 


NOTATION,    DECIMALS,    INTEREST,  <feC.  165 

195.  If  h  be  not  of  the  form  2^5**,  the  fraction  can- 
not  be  reduced  to  a  terminating  decimal. 

For  here  no  factor,  by  which  we  conld  multiply 
both  numerator  and  denominator,  will  make  the  de^ 
nominator  a  power  of  10;  since  all  powers  of  10  con- 
tain only  factors  2  and  5,  whereas  the  denominator 
here  contains  some  factor  different  from  these. 

In  snch  a  case  it  may  be  shewn  that  the  figures  of 
the  decimal  will  recur,  and  the  No.  of  figures  in  the 
period  will  be  less  than  6>. 

196.  To  find  the  Amount  of  a  giveoi  sum,  in  any 
given  time^  at  Simple  Interest, 

Let  P  be  the  principal  in  jpounds^  n  the  length  of 
time  in  years^  r  the  interest  of  £1  for  1  year ;  then  the 
interest  of  P  pounds  for  1  year  will  be  jPr,  and  for 
n  years,  will  be  Pm^  which  is  the  whole  interest  re- 
quired ;  and  the  Amount^  M=P+Prn—P  {l-\-rn). 

If  J[f=2P,  or  the  original  sum  has  doubled  itself, 
we  have  2P  =  P  (1  +  rn)^  and  ^  =  1  -i-  r,  ^=1  -j-  n. 

Thus  at  4  per  cent.,  since  here  we  should  have  r  -  y|^,  and 
.*.  n  =  -^J-  =  25,  it  appears  that  any  given  sum  will  double  itself  in 
25  years ;  but  to  have  doubled  itself  in  15  years,  it  should  be  put 
to  interest  at  6f  per  cent.,  since  then  we  should  have  n  =  15.  and 
.-.  r  =  yV,  andl00r  =  6f. 

CoR.  Hence  the  Simp,  Int.  on  any  sum,  is  propor- 
tional, (i)  to  the  Principal^  when  the  Eate  and  Time 
are  given,  (ii)  to  the  Rate^  when  the  Principal  and 
Time  are  given,  (iii)  to  the  Time^  when  the  Principal 
and  Eate  are  given  {Arithmetic^  96)  ;  but  the  Amount 
only  in  the  first  case. 

197.  To  find  the  Amoimt  of  a  given  Sum,  in  any 
given  time,  at  Compound  Interest, 

Let  P,  n  denote,  as  before,  the  Principal  and  Time ; 
R  the  amount  of  £1  with  its  interest  for  1  year=:l+r; 


II 


156  NOTATION,    DECIMALS,    INTEREST,  AC. 

then  PR  will  be  the  amount  of  £P  with  interest  for 
1  year,  and  this  becomes  \\\QPrincipal  for  the  2nd  year: 
.\PE  X  R^PR  will  be  the  amount  of  £P  for  2  years, 
and  this  becomes  the  Principal  for  the  3rd  year: 
.-.  PR'  xR  =  PR'  will  be  the  amount  of  £P  for  3 
years,  &c.  hence  J}£=PR^=P  (l+r)%  the  amount  of 
£P  for  n  years  :  and  the  interest =P7?^-P«P(^'»-1). 

Coi^.  Hence  the  Com/p,  Int,  on  any  sum,  as  also  the 
Amount^  is  proportional  to  the  Pri7icijpal^  when  the 
Rate  and  Time  are  given  ;  but  the  corresponding  state- 
ment will  not  hold  good,  for  the  other  cases  of  (196  Cor.). 

198.  To  find  the  present  Value  and  Discount  on  any 
Bum  for  a  given  time^  (i)  at  Simple  (ii)  at  Compound 
Interest, 

Let  Y  represent  the  present  value,  D  the  discount,  of 
a  sum  P  due  at  the  end  of  ti  years ;  then,  since  Fis  the 
sum,  which  at  Int.  for  the  given  time  will  amount  to  P, 
we  have  (i)  P=  F(l+m),  (ii)  P—  F(l+r)^  ;  hence 

(i)  F=:  -^,  and  Z>=P~  F=:-^,  (ii)  F=  -^. 
^  1+m'  l+m' ^  ^         (l+r)'* 

Ex.  1.  What  sum  will  in  9  months  amount  to  £600,  at  5  per 

cent,  per  annum.  Simple  Interest  ? 

Here  Jf  =  600,  r  =  y^^  =  .05,  n  =  J  =  .75,  to  find  P: 

7>        ^  600  600        ^^„^    ^^ 

.'.  P=  T =  , ^ — ^^  =  Tl^^zrz  =  £5^  8  65  3^  nearly. 

1  +  r/i      1  +  .05  X  ./5      1.0375  ^ 

Ex.  2.  In  what  time  will  £91  13«  4^^  amount  to  £100  at  3  per 
cent.,  Simple  Interest  ? 

Hero  P=  91f,  r  =  j?^,  if  =  100,  to  find  n : 
.-.  100  =  91f  (1  +  y3^7i),  whence  n  =  -V/  =  SjV  years. 
Ex.  3.  Find  the  Comp.  Int.  on  £275  for  3  y6ars  at  5  per  cent. 
Hero  P  =  275,  n  =  3,  P  =  1.05,  to  find  M: 

\  M=  275  X  (1.05)»  =  £318  6«  ll}r7, 
and  Interest  -  IT-  P  =  £43  6a  11  Jc?. 


MISCELLANEOUS  EXAMPLES :  Part  L 


1.  Multiply  a"  -  lax  -V  ^Ix  by  V  +  ax, 

2.  Divide  '6x'^  +  4aZ?.c-  -  (jo^lrx  -  4a^5'  by  2a5  +  x, 

3.  If  a;  =  1,  y  =  -  2,  5  =  3,  find  the  value  of 

Zx'  -  2xy  +  5?/^  +  53-  +  2y2  +  2xz 
4x^  +  2x1/  +  32/'^  +  25''  +  yz-  xz ' 

4    Ecducc— ^'-^^--  and  ^*  ^  '^V'  ^  ^^ 

a  (inr  +  rr)  -  man  x*  +  2x^y  +  Sx'y^  +  2xy^  +  y* 

5.  Extract  the  square  roots  of  1//^  and  GG.455104. 

r    c-       rr    (H-^)-i(r-U)        _      1  1  aj   4-3 

6.  Simplify  ^j^--^^--.^  and  ^^  -  2^^  -  ^^^. 

7.  Sum  the  a.  p.  7  +  8  J  +  &c.  to  8  and  to  n  terms. 

8.  Insert  an  h.  mean  between  1^  and  1}. 

9.  Reduce  to  their  simplest  forms  Vl^S,  V98a^c,  f  V^lyV 

10.  Expand  (1  -  2a!)  2  to  five  terms. 

11.  (i)     H^^-^)-i(^^-^)  =  H  (ii)    ic  +  7  =  V5^?TT9 
(iii)  ix-iy=^l                                    )   (iv)  ic^-f7/  =  13) 

G  (a;  +  2/)  -  3  (.2J  -  ?/)  =  13  (a;  - 1)  S  a;y  =    6  ^ 

12.  A  certain  fraction  becomes  1  when  3  is  added  to  the  num' 
and  i  when  2  is  added  to  the  den' :  find  it. 


13.  Write  down  the  square  of  1  +  2a;  -  aj'  -  |a;'. 

14.  Divide  51a;y  +  10a;*-48ajV-152/*-f4aJ2/'  by  4x7/--6x^  +  Zy\ 

15.  Find  the  value  of  x*  -2a  (a  -l))x^  +  (a^  +  P)  (a-h)  x-  a'l\ 

when  a=lj  5  =  -  2,  a;  =  3. 
IG.  Find  the  g.  c.  m.  of 

ox^  -  x""?/  -  2y^  and  10a;*  +  15a;V  -  lOa;^'  -  loxif. 

17.  Extract  the  cube  roots  of  1953125  and  5. 

_  o    c-      rr   2  (a;^  -  J)      ,       -J  cc^  +  Sa'a?  +  Sax^  +  a;'        (a  +  xY 

18.  Simphfy  — > ~  +  i  and ^ s-  -^^ ^—> 

19.  Sum  the  g.  p.  3  -  1  +  &c.  to  5  terms  and  ad  infinitum. 

20.  Simplify  {{aH'h^'^^Y''  2iTi^x-^y"^z^{xyz'^. 

21.  Expand  to  five  terms  g — ^ — -. 

-  V  <z  -  3a; 


li  MISCELLANEOUS   EXAMPLES. 

22.  Express  ZOOO  {quaternary)  in  the  quinary  scale,  and  3000 

{quinary)  in  the  quaternary,  and  all  four  in  the  septenary. 
^^    ,.,  3x-2     2\-Zx     Ga;  +  13  ....    ^       ^      1 

23.(0,^^ — 5-=-Tir  oo?-=i-i 

(iii)li-5aa;-l)  =  2-f(y4-l)  ) 

fa;  +  8  -  Jr  (y  -  5)  =  11a;  -  3  J-  (3a;  -  2)  | 
21.  A  can  do  a  piece  of  work  in  10^  days,  which  A  and  B  can  do 
together  in  5}  days  :  how  long  would  B  take  to  do  it  alone. 


25.  Find  the  product  of  a;-  -  a,  x'  -  a"x  +  «,  and  x^  +  a^a;  +  a. 
2G.  Divide  Ja;=^  -  \x^  +  -ya;^'  -  -^-a;'  -  -^-a;  +  27  by  Ix"  -  a;  +  3. 

27.  If  a;  =  1,  2/  =  -  2,  2  =  3,  find  the  value  of 

\\.^-\\y-\{^-'^^^)\\ 

28.  Reduce  — f  and   .>^.  ..  ., — ,^  ,  .. — ,,..  .  ., — r--r-„. 

on    c-      vr   l-ill-ia-a')}        ,      :^  +  2  2-a;  x 

20.  Simplify       ^    1-7-71 — fr  ana  j— — --   +  -^^-r—T^  "  -?-?• 
•^  1-J  {l-^(l-a;)t  2(a;+l)      2  (a;-l)      a;^  +  l 

SO.  Find  the  square  roots  of  19321,  1.9321,  and  19.321. 

31.  Obtain  a  fourth  proportional  to  ?,  J,  |-,  and  a  mean  propor- 
tional to  .017  and  .153. 

32.  Sum  the  g.  p.  |  -  f  +  &c.  to  71  terms  and  ad  infinitum, 

33.  Expand  {ax  -  x")  ^  to  five  terms. 

34.  In  how  many  ways  may  a  sum  of  40  guineas  be  paid  in 
dollars  (45  6d)  and  doubloons  (13s)  ?  and  how  may  it  be  paid 
with  fewest  coins  ? 

35.(i)^-^i?.S7i-?n^ 

(n)ix-12  =  ly.8  )  .^.J:^^^^^^, 

i{x+y)+lx=l{2y-x')  +  Z5]         ^^x  +  2     x       '^' 
30.  A  can  correct  70  pages  for  the  press  in  1  ^  hr,  B  can  correct 
150  pages  in  2 J  hrs  :  liow  long  will  they  be  in  correcting  425 
pages  jointly  ? 

37.  IMultiply  {a  +  h  +  c)  {a  +  h  -  c)  hy  {a  -  1)  +  c)  {h  +  c  -  a), 

38.  Divide  1  -  ^a;  by  1  -  ^a;  -  Ix"^  to  five  terms. 

30.  If  «  =  -  a;  =  i,  6=0,  find  the  numerical  value  of 
"x*  -{a~l)x'  +  {a  -  h)  ¥x  -  l\ 
2x^  -  a;'*  +  a;  +  1 


40.  Reduce  to  its  lowest  terms 


2a;'  +  3a;^  +  3a;  +  1* 


AUSCELLANEOUS   EXAMPLES.  iU 

41.  Find  the  cube  roots  of  2C85G10  and  J. 

42.  Simplify  the  fraction  i^^^^^^^^-^ 

43.  Expand  (a*  -  4a'a;^)T  to  five  terms. 

2a  s  /  0        ^  Sx  *  /80y* 

44.  Reduce  to  their  simplest  forms  -^  ,i/  —  and  ~  a/  Trp^ . 

45.  Sum  the  a.  p.  ^  +  f  +  &c.  to  31  and  to  n-2  terms. 

4G.  Transform  1828  into  the  septenary  scale,  and  square  it ;  re- 
duce the  result  to  the  nonary,  and  extract  the  square  root  •, 
and  express  the  latter  two  results  in  the  denary. 

47.  (i)  Zx-i(x-l^)=:d-i{ox-7) 

(ii)   X-  ij  -z=    Gl  (iii)  a(:x  +  y) -I)  (x- y)  =  2a^) 

Z?/-x-z=::12[  (a'-h-)(x-y)=4a'bS 

7z  -y  -x  =  24:  \ 

48.  Two  men  can  do  a  piece  of  work  in  12  days,  and  one  of  them 
can  do  half  as  much  again  in  24  days ;  in  what  time  could 
the  other  do  a  third  as  much  again  ? 


40.  Simvrify  i  {la -(h-a)\-ilib- la)- I  {a.~-l(h-ia)]]. 

50.  If  a  =  1,  5  =  3,  c  =  5,  fmd  the  numerical  value  of 

|«_(5_c)p  +  |5_(c-a)p+  \c-(a-h)}\ 

51.  Expand  and  simplify  the  quantities  in  the  preceding  question. 

52.  Find  the  g.  c.  m.  of 

T.i;'-2a;'7^-G3;r2/''  +  18?/'  and  6x'-Zxhj-AZxY^  +  27xy^-18y\ 

53.  Extract  the  square  roots  of  IIIOOIG  and  9  +  2  Vl4. 

54.  Simplify 


a+h+  — I  -i-  (a  +  5+  -7-    and  {a-h+  —  , 
a         \  0  I  \  a+b 


a-hj 


55.  Sum  .2  +  .02  +  .002  to  n  terms  and  ad  ivfmitum, 

5G.  IIow  many  terms  of  the  series  17,  15,  &c.  will  make  72  ? 

57.  Expand  (a*  -  bx)"^  to  five  terms. 

58.  IIow  many  different  throws  can  be  made  with  two  dice  ? 
UO.  (i)  _A_  =  8-2  ^'^'^  "■  ^^  ^"^  5^  +  7y  =  43  ) 


ic+1  \x+Z)  llaj  +  9y=G9) 

(iii)  x-y  —  xy-  =  G  =  2xy, 

tV  person  bought  cloth  for  £12:  if  he  had  bought  one  yard 
)ss  for  the  same  money,  each  yard  would  have  cost  him  \% 
lore ;  how  many  yards  did  he  buy  ? 


IV  SUSCELLANEOUS   EXAMPLES. 

CI.  Multiply  2y  +  Zx^i/-  x^  by  7x^  -  Syi 
C2.  Divide  x'  +  4x  +  S  by  a;^  -  2^  +  3. 


C3.  If  a  =  1,  &  =  2,  c  =  3,  find  the  value  of  Va  (6'  +  <z<;)  -  |  J'c. 

G4.  Find  the  c.  c.  m.  of  a^  (h*  -  I'c')  and  l^  {ah  +  acf, 

G5.  Obtain  the  fourth  root  of  Ux^  {x  -  2)  -  8j;^  (a?"^  -  3)  +  1. 

a;  +  l       x-l  l-3.tj 

C6.  Simplify  ^^--j  -  ^^-^  -  — ^-^^ . 

67.  Find  the  g.  mean  between  12|  and  13,  to  3  places  of  decimals. 

d^.  Expand  f-^ A^  to  five  terms. 

'  \a^x  -  ax^j 

CO.  What  number  is  that,  which  is  just  as  much  below  35  as  its 

half  is  above  its  third  part  1 

70.  Convert  297  to  radix  11 :  square  and  cube  it  in  that  scale, 
extract  the  roots,  and  reconvert  them  to  the  common  scale. 

71.  (i)  \  {Zx  +  5)  - 1  (21  +  ic)  =  39  -  Sa-. 

(ii)  2a;'  +  a;  =  28.  (iii)  2a;  -  9?/  +  2  =  0  =  3a;  -  12^/  +  21. 

72.  A  and  B  can  reap  a  field  in  lOhrs,  A  and  (7  in  12  hrs,  B  and  C 
in  15  hrs:  in  what  time  can  ihGy  ^'^'^^ jointly  ixxidi separately  f 


73.  Obtain  the  quotient  of  C  V^'  -  ^G  Va;"*  by  Ijx  -  2  V^~'. 

74.  If  a;  =  I,  and  a;  +  y=a;  +  y  +  s  =  0,  find  the  value  of 

^^    ^   ,  x(x^  -^  y^)  (x-y)  -  a;*  +  So;*  +  a;  4-  3 

75.  Reduce  .  .       K-rr-^  >,         ^  ^'^^ j—q q 

{x^  -  2/  )  (.^  +2/       ^2/)  cc^  -  oa;  +  3 

7G.  Add  together  7vG3  +  2  V252  +  11  V28. 

77.  Find  V3.14159,  and  the  fourth  root  of  a;*-  ^o;  +  fa;'  +  iV  -  2^*- 

78.  Shew  by  the  Bin.  Theor.  that  V^  =  1  +  i  -  i  +  iV  "  tI?  +  ^c. 

79.  Sum  the  a.  p.  f  +  2  +  &c.  to  9  and  to  n  terms. 

80.  Form  the  equation  whose  roots  arc  2,  -  2,  1  +  -^5,  1  -  ^5. 

81.  What  number  is  that  which  is  the  same  multiple  of  7,  that  its 
excess  above  20  is  of  its  defect  from  30  ? 

82.  How  many  different  arrangements  can  be  made  of  the  letters 
of  the  word  Notogorod?  IIow  many  with  two  o's  at  the  be- 
ginning and  two  at  the  end  ? 

83.  (i)i(7a;  +  5)-f  (.c  +  4)  +  G  =  #  (;i'+ 3) 

(ii)  X  +  y-8  =  0=  i  (x-y)  +  Ux-\y  +  2) 
(iii)  a;  +  V5a;  +  10  =  8. 


MISCELLANEOUS   EXAMPLES. 


84.  Out  of  £5000,  a  person  leaves  £20  to  an  old  servant,  and  the 
remainder  among  three  societies,  A^  i?,  and  6',  so  that  I)  may 
have  twice  as  much  as  C,  and  A  three  times  as  much  as  B: 
how  much  docs  each  receive  ? 


85.  Multiply  V^'  -^  1  +  ^^  by  V'l-'  -  1  +  ^- 

80.  Divide  ^a^  +  ^a'X  -  2x^  hj  \a  +  x. 

87.  If  a  =  1,  5  =  I,  ic  =  7j  2/  =  8,  find  the  numerical  value  of 


^{a-l)  V  {a+x)  yUa-h\fJa7x)i/-  '^^if  -  {a  -  V3  (x+2h)\\ 

88.  Simplify  1^-^  |1-J  (x-i)\  and  \a-^(a-ih)\-^{h-^  {a  +  'il)\. 

89.  Write  down  the  quotient  of  ax~^  +  2>^  by  a"x~^  +  Z>^. 

90.  Find  the  square  root  of  (x  +  x'^)  -2(x'^  -  x'^)  -  1. 

91.  Sum  the  a.  and  g.  p.  |  +  2  +  &c.,  each  to  n  terms.  Can  the 
latter  series  be  summed  ad  infinitum  ? 

92.  Expand  Vl  +  4.c  to  five  terms,  and  square  the  result. 

93.  Find  two  numbers  in  the  ratio  of  1^  :  2f ,  such  that,  when 
increased  by  15,  they  shall  be  in  the  ratio  of  1|  :  2\. 

94.  In  how  many  ways  ma)^£24  IGs  be  paid  in  guineas  and  crowns? 

95.  (i)-^9:c  +  7)-jaj-4(a;-2)}  =3G 
(ii)  05+ 1  :  2/ ::  5  :  3  ) 

|:r-i(5-2/)=3^-i(2^-l)5 

9G.  A  messenger  starts  with  an  errand  at  the  rate  of  3|  miles  an 
hour ;  another  is  sent  half-an-hour  after  to  overtake  him, 
which  he  does  in  2  hours:  at  what  rate  did  he  ride?  Find 
also  in  what  time  he  will  do  it,  if  he  rides  12  miles  an  hour. 


97.  Simplify  |  {x  (x^l)  {x-^-2)  +  x{x-V)  {x-2)]  +  §  {x-l)  x  (aj+1). 

98.  Divide  a'  -  -'/crl''  +  ^al^  +  ^l'  by  a^  +  2ah  +  4^h\ 

99.  Find  the  g.  c.  m.  of  Zx^  +  4;c^  -  3u;  -  4  and  2x'  -  Ix'  +  5. 

100.  Reduce  -^fJQS^         and  ^  ^  V^^. 

{x-  +  V- 2bx)  (bx  +  x')  x-a^y 

101.  Find  the  cubo  root  of  C9.42G531. 

5.  ^2  4  .4  2       3  _2  2 

102.  Multiply  1  +  a**  -a;  i^  -f  a3"  +2?  ^  +  rt^ic  ^  by  oj  3-  ^a  + 1. 


VI  MISCELLANEOUS   EXAMPLES. 

103.  Find  the  common  difference  of  an  a.  p.,  "when  the  first  term 
is  1,  the  last  term  50,  and  the  sum  204. 

104.  If  a  :  h  ::  c  :  d^  shew  that  7  a  +  h  :  oa  —  51 ::  7c  +  d:  3c-  5cZ. 

105.  Divide  100  into  two  parts,  so  that  ^  the  greater  ma}^  be 
greater  than  ^  the  less  by  J  their  difference. 

lOG.  Employ  the   septenary  scale  to  find  the  side  of  a  square 
which  contains  a  million  square  feet. 

107.  (\)l-(x■,^)-un-x)  =  |(x-4)-iJ(x-^) 

2x-l      2x^;^_o        (iii)  3a;-y +  5  =  17  1 

^^'-^  2a;  +  1  ""  2aj  -  1  ~  5  (aj  +  2/  -2)  =  2  (2/  +  s)        [ 

4(.^  +  2/  +  2)  =  3(l-a;  +  32)  J 
108.  A  and  B  engaged  in  trade,  A  with  £275,  B  with  £300  ;  A 
lost  half  as  much  again  as  B^  and  B  had  then  remaining 
half  as  much  again  as  A :  how  much  did  each  lose  ? 


100.  lia-l)  -  x  =  Z  and  «  +  &  +  ic  =  2,  find  the  value  of 
{a-l)\x''-2ax''  +  a''x-(a+  1)1'']. 

110.  Shew  that  (2a  +  5"^)  (25  +  or')  =  {2ah^  +  ah'^y, 

111.  Find  the  l.  c.  m.  of  6a;^  -  13a?  +  C,  Gaj'*  +  5a;-  6,  and  Oa;^  -  4. 

112.  Obtain  the  square  root  of  Ja;*  +  \a*-  \ax(2a^  +  3a;'-4aa;). 

113.  Obtain  -^G  to  four  places,  and  thence  find  -^J,  ^f,  -^1^. 

114.  Siraphfy and  r +    , +  - — r-r 

115.  Square  a  -  25  -  3c  and  2a  -  ihx  -  icx^  +2dx\ 

IIG.  Sum  the  G.  p.  5  +  2  +  &c.  to  7i  terms  and  ad  infinituin, 

117.  The  trinomial  ax"^  +  hx  +  c  becomes  8,  22,  42  respectively, 
when  X  becomes  2,  3,  4 :  what  does  it  become  when  a;  =  -  J  ? 

118.  Expand  Vl-4a;  to  five  terms,  and  obtain  the  same  by  Evol". 

119.  (i)  ^(4a;-  21)  +  3f  +  i  (57  -  Zx)  =  241  -  yV  (5a;-9G)-  11a? 
(ii)lla?^  +  l  =  4(2-a;)' 

(iii)i(3.c-2y  +  l).l(.^_2/)  =  S2/l 
5__3  _  15  I 

~x     2y~2xr  J 

120.  A  and  B  sold  130  ells  of  silk,  of  which  40  were  ^'s  and  90 
i?-s,  for  42  crowns  ;  and  A  sold  for  a  crown  ^  an  cU  more 
than  B  did.     IIow  many  ells  did  each  sell  for  a  crown  ? 


MISCELLANEOUS   EXAMPLES.  Vll 

121.  Write  down  the  quotient  of  IG  -  81fit  by  2  +  3  \la, 

122.  Multiply  «"  +  §(«  +  V)x-\3y  and  a?-l{a-l)x-^  ix^ 

123.  Reduce  to  its  lowest  terms  -^-^ — —-i — . 

Vlx^-bx^  +  4aj-4 

124.  Find  the  l.  c.  m.  cf  a?  ±  ic',  (a  ±  xY,  and  a''  ±  a;'. 

125.  Obtain  the  square  root  of  1^  and  of  12  +  G^3. 

12G.  Simplify  a-ih-c) -\l-{ci-c)\  -{ci-  .;2&  -  (a  -  c)|l,  and 

^,    ^         «  +  c  5  +  c  ic  +  c 

shew  that  - — r—. -  - — —  = --. — . 

{cL  -  0)  (x  -  a)      (a  -  h)  (x  -h)      {x-  a)  {x  -I) 

127.  Sum  the  a.  p.  ^  +  ^  +  <S:c.  to  7  and  to  n  terms. 

128.  How  could  a  sum  of  £24  IG5  be  paid  from  ^  to  i>  with  tho 
use  of  fewest  coins,  if  A  have  only  guineas  and  B  crowns  ? 

120.  Simplify  VS  {a'x-vax'^yiQa'x'  and(V«)*"M(«^^V^r^^*' 

130.  Compare  the  numbers  of  combinations  of  24  different  letters, 
when  taken  7  and  11  together;  and  also  when  the  letters 
a,  Jj  c  occur  in  each  of  such  combinations. 

-.,     ..^Gjj+18     .,     ll-3:c     „        ._     U-x     21-2.it 

131.  0) -^-41—^^  =  5^-48-— ^_ 

(ii)  1 -f  lOy  +  5)  - 1  (T^- 6)  =  10-f,- (3a^  - 10  4- 72/ ) 
J(12-a'):5.iJ-l(14  +  2/)::l:8  J 

On)  5^  -  _-^   =  2x  .  — ^  . 

132.  A  party  at  a  tavern  had  a  bill  of  £4  to  pay  between  them, 
but,  two  having  sneaked  off,  those  who  remained  had  each 
28  more  to  pay  :  how  many  were  there  at  first  ? 


133.  Shew  that  {ac  ±  Id)''  +  {ad  ±  lcy=  (a^  +  1-)  (c-  +  cV\  and  ex- 
emplify this  identity  when  a=l=-d^l)  =  2=—  c. 

134.  Obtain  the  product  of  x+2^/xhj  +  2^/yhjx-2\'x-y->r2jy. 

135.  Divide  x"^  -  (ctr  -  h  -  c)  x'  -  (l -c)  ax  +  he  by  x'^  -  ax  -i-  c. 

13G.  Reduce  ?f:^^^\%  and      ^^  "  ^^'^'  "  ^^' 


10(6=^  -  Oay  -  9y'  Ox^  +  53.r=  -  9a;  - 18  * 

137.  Find  the  l.  c.  m.  of  mhi-mn^,  m"  +  w?z-2;i-,  and  m--r7in-2n\ 

138.  Obtain  the  square  root  ofa^-2a^  +  Za^-2a'^^  +  1. 

139.  l^  a  :  h  : :  c  :  d,  express  (h  +  d)  (c  +  d)  in  terms  of  a,  5,  c, 

140.  Find  -^24,  and  thence  deduce  the  values  of 

5  V2  2V3  +  V2      1  -f  V21G 


via  MISCELLANEOUS  EXAMPLES. 

141.  Insert  two  a.  and  two  ii.  means  between  1  and  3. 

142.  Expand  (1  -  4aj)"^  and  (1  -  4:r)"^  to  five    terms  ;   and   shew 
that  the  former  sericSj  when  squared,  coincides  with  the  latter. 

143.  (i)  ix-i(x-2)  =  i  {x^l.(2i-x)\  -  \(x-5) 

X        x-d  _x  +  1      x-S 


.....  7a;  +  1       80  fx-  i\ 


144.  A  farmer  bought  5  oxen  and  12  sheep  for  £G3,  and  for  £90 
could  have  bought  four  more  oxen  than  he  could  have 
bought  sheep  for  £9  :  what  did  he  pay  for  each  ? 


145.  Find  the  continued  product  of  {x  +  a)  (x  +  h)  (a-2x)  (b-x), 
14G.  AYrite  down  the  square  and  fourth  powers  of  a  -  \^Jax  -  2x, 

1A7    c-      IT     (x'-4x)(x'-4y       .    (a^-l)K-l) 
147.  Simplify  1__— A— and    ' 


{x''-2xy'  (a+  ly  (a^-af 

148.  Reduce  to  its  lowest  terms  r-^ — j,o    o  2    o"i — u-i — T7{ —  • 

Sa^  +  46^  +  3c^-  Hah  -  Sbc + lOac 

149.  Extract  V-Ol  to  four  places  of  decimals. 

150.  Obtain  the  square  root  of  (x  +  ly  -  4^/.r  (x  -  ^^  x  +  1, 

151.  Determine  which  is  the  greater  ^2  -f-  \/3  or  ^Jo  -f- 1/5, 

152.  Sum  the  g.  r.  -^  +  ^  +  &c.  to  71  terms  and  ad  injinitum, 

153.  Given  —  1  to  be  a  root  of  the  equation  ic*  -  Ix'  -  6a;  =  0,  find 
the  other  three  roots. 

154.  In  how  many  different  ways  could  a  farmer  lay  out  a  sum  of 
£G3j  in  buying  sheep  and  oxen  at  30s  and  £9  respectively  ? 

155.  (i)  a  {x  -J))  z=  I  (a  -  x)  -  (a  +  l)x 

3  5  4  (iii)  2a;^  +  3x2/ =  26 ) 

^"^l-Sa;""  l-'ox'^2x~l~  Zif^2xy^Z^\ 

156.  A  and  B  can  do  a  piece  of  work  together  in  4  days  :  A  works 
alone  for  two  days,  and  then  they  finish  it  in  2^  days  more : 
in  what  time  could  they  have  done  it  separately  ? 


157.  Fmd  the  value  of  ^\/\^l  +  ^^^~^+  VS^=^^  +  4J- 
when  a  =  J,  5  =  ]. 


MISCELLANEOUS    EXAMPLES.  IX 

158.  Divide  a^  +  2ahi  +  h^  by  a^  +  2a^  h^  +  h. 

159.  Find  the  g.  c.  m.  oi:x'  +  7x''  +  7x'-~15x  and  aj'-22;'-13a;+110. 

IGO.  Simplify 5 -:  -  -^ — 7  and  :, ,    \,  .-,     \  .. 

IGl.  Multiply  together  a;  -  1  +  V-)  a;  +  2  +  V3,  a;  --  1  -  V2,  and 

aj  +  2  -  V^- 
162.  Find  the  7*^^  term  of  S  +  SI  +  CJ  +  Ac,  and  its  sum  to  IG  terms. 

1G3.  If  a  :  Z> ::  &  :  c ::  c  :  tZ,   shcw  that  a  :  I) ::  ^ <z  :  ^ cZ  J    and  cxprcss 

(a  +  Z))  (c  +  <Z)  in  terms  of  h  and  c. 

164.  Find  the  least  number  which,  when  divided  by  30  and  50, 

shall  leave  remainders  16  and  27  ^cspcctivel3^ 

3. 

165.  Expand  (1  +  2x'^)  ^  and  (a  +  25)'-^,  each  to  five  terms. 

1G6.  Express  a  million  in  the  senary  scale,  extract  its  square  and 
cube  roots  in  that  scale,  and  reduce  the  results  to  the  denary. 

167.  (i)  f(aj-5)-tV(aj^l3i)  =  5-H7-^) 

,..,  x-ay     -      ax  +  1/     ...^  3a;  +  8      5  (12  -  a*) 
^  ^      b  c        ^^aj-4        2^  +  3 

168.  If  ^'s  money  were  increased  by  half  of  i?'s,  it  would  amount 
to  £54 ;  and,  if  ^'s  present  sum  were  trebled,  it  would  ex- 
ceed three  times  the  difference  of  their  original  sums  by  £6. 
What  had  each  at  first  ? 


169.  Write  down  the  expression  for  the  product  of  the  square  root 
of  the  sum  of  the  cubes  of  the  square  roots  of  a  and  b,  by  the 
square  of  the  cube  root  of  the  sum  of  their  squares :  and  find 
its  value  approximately,  when  a -4,  b  -1, 

170.  Multiply  x'^  +  2x'^  y^  +  3?/  by  x'^  -  2x'^'i/^  +  y. 
la.  Simplify^  (^-j    ^      ,  and  reduce 2-^-^^-,--3-. 

172.  Obtain  the  square  root  of  1  -  ayr  -  -^-a^x  +  2a^x'*  +  Aa^x'^. 

1-0    v.-    I  x»  ^«  +  Z>         b  ,    ^1       2.^+1         4a,' +  5 

1  /  3.  Fmd  tnc  sum  of 7.  and  of  1  +  pj- ^-  -  -pr- — ^ 

x-a     x-b-  2  (a;  -  1)      2  (ic+ 1) 

174.  Extract  -^15,  and  thence  obtain  the  square  roots  of  |,  f.  2f^ 

412  ■ 

1x3. 

175.  Sum  the  a.  p.  13  +  11^  +  &c.  to  5  and  to  n  terms,  beginning 
in  each  case  with  the  ninth. 

8* 


X  MISCELLANKOrS   EXAMPLES. 

17G.  If  a;  =c  ^-^,  y  =  ^4.  ^^^  the  talue  of  x^ -^  xy  ^  y\ 

177.  Expand  (1  +  -^.x) ^  to  five  tcrmSj  and  obtain  from  the  result 
tlio  scries  for  (1  +  -^.c)  *. 

178.  Find  three  numbers  in  the  proportion  of  |,  §,  f,  the  Bum  of 
whose  squares  is  724. 

x-\ 

179.  (i)  6a;-a:4a?-6::3aj+  &:2aj  +  «       (ii)  3(a;-J) p:  =  5 

ic  +  2 

(iii)  {x  +  5)'^  +  (?/  +  C)=  =  2  (a^z/  --  24),  y  =  cc  +  1. 

180.  A  docs  §  of  a  piece  of  work  in  G  days,  when  B  comes  to  help 
him  ;  they  worlc  at  it  together  for  ^  of  a  day,  and  then  jB  by 
Ijimself  just  finished  it  by  the  end  of  the  day:  in  what  time 
could  they  have  each  done  it  separately  ? 


181.  Find  the  continued  product  of  a  +  ic,  a  +  \y,  a^\z\  and 
deduce  from  the  result  the  value  of  {a  +  lif, 

182.  Multiply  \x  +  3^"^  -:^a^  by  1x  -  a'M  ^  \a^, 
.183.  Simplify  x-\\Q.\-x)-\{^\-x)-l  (H  -a;-2i)}. 

T  o  I    T.   1       X    -x    1         X  .L         ic"  -  oj^  +  3.7?^  -  2.C  +  2 

184.  Kcduce  to  its  lowest  terms 7 — z-;. — -z — . 

x^  —  b7j'  +  Oil'  —  5 

185.  Find  the  l.  c.  m.  of  ax^  -  a%,  ax^  -  1,  and  «ic'  +  1. 
18G.  Extract  the  square  root  of  a^^^  ^  +  \a  ''V  -  a'l)  +  2db  \ 

187.  Find  the  sum  of  77-- r-r  + —-  -  - — - — r^. 

2  (x  -  \y      2  (a;  -  1)      2  {X-  +  1) 


188.  Multiply  together  3  ^8,  2  ^G,  V^^.  V^O  ;  and  find  V2+Tv7. 

189.  Sum  the  g.  r.  G  -  2  +  &c.  to  7  and  to  n  terms. 

190.  A  watch  which  is  10'  too  fast  at  noon  on  Monday  gains  3'  10" 
daily :  what  will  be  the  time  by  it  at  7h  12'  a.m.  of  the  fol- 
lowing Saturday  ? 

101.  (i)  Jr  (3x  +  I)  -  ^  (4x  -  C§)  =  i  (5.r  -  C) 

(ii)  5x+4y=38i  +  i  {Zx-y),  x^5^\  -  J  )i  («+y)  -  {{x-y)]. 
,„.,     2  3  5 

192.  A  man  and  his  wife  w^ould  empty  a  cask  of  beer  in  IG  days  ; 
after  diinking  together  G  days,  the  woman  alone  drank  for 
9  days  more,  and  then  there  were  4  gallons  remaining,  and 
she  had  drank  altogether  SJ  gallons.  Find  the  number  of 
gallons  in  the  cask  at  first.  . 


MISCELLANEOUS   EXAMPLES.  XI 


193.  If  Aa  =  5h=  1,  find  the  yalue  of 

iC^"  x""'  1  1 


1  (J  +  a-V);*-vQ  u  +  «"^  -a  +  «6-)'*}]. 


194.  Find  the  sum  of  -      .  ^        „     i   • 

iC' -  1        iC"  +  1        iC'*  -  1        .'C"  +  1 

195.  Simplify  the  surd  expressions 

3V2  +  2V3      3V2-V3      3_V^+_2Vi 

196.  Reduce  7-^ »//  .. ^^  and 


(m'^  -  a^)  (m^  -  am  -  2a^)  a'  -  2a\t  -  «ai*  +  2a;'' 

19T.  Find  the  l.  c.  m.  of  3a;-  -2:c-l  and  4.i^-2a;-  -  3a;  +  1. 

198.  Sum  the  scries  3  -2  +  IJ—  &c.  to  ti terms  and  ad  ivfinitum, 

199.  Prove  that  the  sum  of  any  number,  ti,  of  consecutive  odd 
numbers,  beginning  with  unity,  is  a  square  number. 

200.  Given  if  ^  a^  -  a;^,  and  when  x  -  ^d^  -h^,  ai/=  h',  find  the 
value  of  X  when  y  =  |&. 

201.  A  person  distributed  £2  Is  Sd  among  some  poor  people, 
giving  9^d  to  each  man  and  6^d  to  each  woman :  how  many 
men  were  there,  it  being  known  that  the  whole  number  was 
a  multiple  of  10  ? 

202.  Expand  (1  +  VO"  ^^  ^^'^  terms,  and  obtain  from  the  result 
by  Evolution  the  series  for  (1  +  yx)"^. 

203.  (i)  i  {I  +  t(a;  +  2)f  -f  {IJ-  (l^-o^)}  =  h%  

(ii)  ahx"^  -{a  +  h)x+l==0        (iii)  i(x+y)  =x-y=  '\/x  +  2y-l. 

204.  A  and  B  lay  out  equal  sums  in  trade ;  A  gains  £100,  and  B 
loses  so  much,  that  his  money  is  now  only  f  of  ^'s ;  but  if 
each  gave  the  other  J  of  his  present  sum,  ^s  loss  would 
be  diminished  by  one  half.  What  had  each  at  first,  and 
what  would  ^'s  gain  be  now  ? 


205.  Shew  that  \  (x^  +  y^)  +  z^-^xy  +  xz-yz  and  (y  ~  zy  become 

identical  when  -x  =  y  =  a, 
20G.  Divide  mpx^  +  (mq  -  np)  x^  -  {mr  +  nq)  x  +  nr  by  mx  -  n 

207.  Multiply  <*^  +  a'^"  +  2 - a^  +  ^"3  by  cC^-a^  +  1. 

208.  Reduce  to  its  lowest  terms — — — — 3  • 

x^  -ax  +  rt-a;-a^ 

1  -  2aj  1  +  a;  1 

209.  Obtain  the  sum  of  V 


Zif-x^X)      2(a;-  +  l)      6(a;+l)' 


Xii  MISCELLANEOUS   EXAMPLES. 

210.  Find  the  square  root  of  40.14290404  and  the  cube  root  of 
8242408. 

211.  The  Z'^  and  IS^'^  terms  of  an  a.  p.  arc  3  and  l:  find  the  14* 
term,  and  the  sum  of  20  terms. 

212.  Simph'fy  the  sUrd  expression  {ah-^ .  V«^^ .  Vab* .  VaJ}}'^. 

213.  The  forc-whcel  of  a  carriage  makes  G  revolutions  more  than 
the  hind  wheel  in  120  yards,  and  the  circumference  of  one  is 
a  yard  less  than  that  of  the  other  :  find  that  of  each. 

214.  Transform  1000000  from  the  quinary  to  the  septenary  scale; 
and  extract  its  square  and  cube  roots  in  the  latter. 

215.  (i)i(x-l)(x--2)  =  (x-2^(x-l^ 

(ii)  2x  +  S?/  =  5  =  -(21/  +  Sx)        (iii)  xU  xy=a'',  y''+x2/-^^h\ 

216.  Find  the  time  in  which  A  and  B  can  do  together  a  piece 
of  work,  which  they  can  do  separately  in  m  and  n  days. 
How  long  must  A  work  to  do  what  B  can  in  m  days? 

217.  Find  the  difierence  between  (n  +  2)  (n  +  3)  (n  +  4)  and 
24  ^7^- 1(7.-1)}  {n~IO^-2)}   {n-^(n-li)\. 

218.  Divide  a  +  &^  +  c'  -  3  V^6V  by  a^  ^  h^  +  c, 

219.  Fmd  the  sum  of + ^        ^ 


x-h     X  -^  a     (x  -  a)  {x  -  b)' 

220.  Find  the  l.  c.  m.  of  x^+  xhj  +  xy^  +  y^  and  x^  -  xhj  +  xy^  -  y\ 

221.  Obtain  ^\0^  and  thence  derive  the  values  of  |  ^f,  ^4 J,  ^'24, 
(V5  +  v2)-*-(V-5-V2),  and  (VS-V^)  -  (^  V2-2V5).  " 

222.  Sum  (10-^  +  2-»  +  (2f)-^  +  &c.  to  n  terms  and  ad  injinitxm, 

223.  Expand  {a"  +  2.r-)"^  and  (2^-3ir)-2  each  to  five  terms. 

224.  A  servant  agrees  with  a  master  for  12  months,  on  the  con- 
dition of  receiving  a  farthing  the  first  month,  a  penny  tho 
second,  fourpence  the  third,  and  so  on :  what  would  his 
wages  amount  to  in  the  course  of  the  year  ? 

225.  Given  two  roots  of  the  equation  ic*  +  4aj  =  bx^  to  be  1  and  -2, 
find  the  other  three  roots. 

226.  A  person  changed  a  sovereign  for  25  pieces  of  foreign  coin, 
some  of  them  going  30  to  the  £,  the  others  15 :  how  many 
did  he  get  of  each  ? 

227.  (i)  2ax^  +  {a-2)x-\  -  0  (ii)  ax^\^hy^\=ay-^hx 

...     X        aj  +  2_8j;-13 


MISCELLANEOUS  EXAIklPLES.  XlJl 

228.  Find  the  time  in  which  A,  B,  and  C'can  together  do  a  piece 
of  work,  which  A  can  do  in  m  daj^s,  B  in  n  days,  and  C  in 
i  (m  +  n)  days. 

229.  Divide  5?/*  +  lay^ -^ -Y^'-aY'  +  -^2/  +  W  ^7  iv""  +  oay-la\ 

230.  Obtain  the  products  of  ^x^  +  a  V^;'  +  a"^  (i)  by  ^x^-aijx^^-a^^ 

(ii)  by  ^x^  +  aXjx^-a^^  (iii)  by  ^x^'-aXjx^-a'^. 

231.  Find  the  g.  c.  y..  of 

3a^-a"6^-26*  and  lOa^  +  15a^5-10a-Z>^-15ai^ 

232.  FindtheL.c.M.ofii'^-3a,-^+3a;-l,  a^^-rc^-ic+l,  x'-2x^^2x-X 

anda;*-2ic«  +  2a;'-2.'c  +  1. 

238.  SimpHfvr  _-i^  and  ^-^2-  ^. 

234.  Extract  the  fourth  root  of 

2  0  4  li>      8  5_    1  2  in 

Y^x'^'  -^x^'y^  +  -Y-x^'y^  -2!J0x^y'^'  +  G25?/"^". 

235.  Sum  IG^  +  14|  +  13  +  &c.  to  11  terms,  and  |  +  |+|;-  +  &c.  to 
n  terms  and  ad  inf, ;  and  insert  8  h.  means  between  1  and  2. 

236.  Given  y"^  '-¥(ax  ^  «,  and  when  x  =  h.y  =  a,  find  the  value 
of  y  when  x  =  3a» 

237.  Four  places  lie  in  the  order  of  the  letters  A,  B^  (7,  JD,  A  is 
distant  from  I)  34  miles,  and  the  distance  from  -4  to  -5  is 
I  of  that  from  C  to  I)  ]  also  J  of  the  distance  from  -4  to  i?  is 
less  than  thrice  the  distance  from  B  to  0  hy  |  of  the  dis- 
tance from  C  to  B.     Find  the  respective  distances. 

238.  If  (l  +  xy  =  l+  Atx  +  &c.,  and  (I  +  x)""  =1  +  Ba  -^  &c., 
shew,  by  finding  the  actual  values  of  J^i,  i?i,  &c.  that 

A,  +  A^Bi  +  A1B2  +  i?3  =  0. 

239.  ({)  3x^20  =  7-^]Z-^(x-l)\      (ii)  -  +  -  ^  a,  -  +  ^=5 

y  X      y 


(i) 

3x* 

20  = 

7-i 

■13- 

•H^- 

■1)} 

(ii) 

X 

(iii) 

Gy- 

-4x 

5z- 

-  X 

y- 

-22! 

=  1. 

33- 

-7 

Zy. 

-2a; 

240.  If  in  (228)  A  work  for  \  {Zm  -  2i\)  days  and  B  for  i  (37i-2w) 
days,  in  what  time  will  G  finish  the  work  ? 

241.  Write  down  the  quotient  of  x"^  -y^  by  x'^  +  y"^.  and  divide 
jc^  -  2ax'  +  (rr  +  a&  -  Jr)  x  -  a"J>  +  aJi^  by  a;  -  «  +  &. 

242.  If  a  =  16, 5  =  10,  a;  =  5, 2/  =  1,  find  the  value  of  (x-b)  {^a-  h) 
+  V(rtt  -b)  (x  +  y)  and  (a  -  a;)^  -  (&  -  a'^)  -  V  (<^?  -  .c)  {h  +  y). 

243.  Find  the  g.  c.  m.  of  3 00^^3^265 j-'^  +  50r  +  2\  and  OO.i'-53.T-f  4. 


XIV  MISCELLANEOUS   EXAMPLES/ 

244.  Simplify  — — -  and   I3  .  -^^j-^j  x  j-  -  ^^^^  ^. 

245.  Find  ^-G,  and  obtain  by  means  of  it  the  values  of 

V^t,  ^/^^  (V3  -  V2)',  and  (2  V3  +  3  ^2)  +  (3  V3  -  2  V2). 
24G.  Shew  that  V{«'  +  V^^}  +  Vl^'  +  Va^*}  =  (a^  +  P;2. 

247.  Divide  48  into  nine  parts  so  that  each  may  just  exceed  that 
which  precedes  it  by  i. 

248.  Given  the  coefficients  of  the  4*^  and  G'^  terms  of  (1  +  a-)*^"^^ 
equal  to  one  another :  find  n. 

249.  In  the  permutations  of  the  first  eight  letters  of  the  alphabet 
how  many  begin  with  ahl  "^ 

250.  Express  12345G54321  in  the  scale  of  12,  and  extract  its 
square  root  in  that  scale.. 

251.  (i)  f(.r-5)-tV(-r-13J)=15--H19--Ja') 

(ii)  ax  -hj  =  a"" )  (8x  -  3\2  _  4a; -5 

hx-a7j  =  h^\  ^^"-^  [i^'i)   ~  x-i 

252.  Find  the  time  in  which  A,  B.  C  can  together  do  a  piece  of 
work,  which  (i)  A  can  do  in  m  days,  and  B  and  C  together 
in  ^  {m  +  n)  days,  or  (ii)  A  can  do  in  m  days,  A  and  B  in  n, 
and  A  and  6^  in  ^  (m  +  n)  days. 


253.  Find  the  coefficient  of  a;  in  (a;  +  2)  (x  -  6)  (a;  +  10)  (x  -  5),  and 
of  a;-*  in  (1  +  -j^a;  +  Ix^  +  ix^  +  &c.)  x  (1  -  |a;  +  ]a;^  -  -^a;^+&c.). 

254.  Divide  a;  *  +  7/  by  a;'5"  +  y^  and  a;^  -  ma-x^  +  max^  -  a^  by 

a;^  -  ar^, 

255.  Find  the  l.c.m.  of  Gx''  -  Ux"  +  5a;  -  3  and  9a;''  -  9a;'  +  5a;  -  2^ 

25G.  Simphfy  :{  ^  wi     o  ^?  and  reduce  -^ r^ .. 

l-^(l  +  2.r)'  a^+a^'b-a-h 

257.  Find  the  sum  of 

a  ac  a     h-2c       ^    2-2a;-  1 

-,  and 


b      b{b  +  c)      b       b-c'  ^/([^^y     VE^'' 

258.  A  walks  at  the  rate  of  3  miles  an  hour,  ^  starts  2  hours  after 
him  at  4  miles  an  hour :  how  many  miles  will  A  have  walk- 
ed before  B  overtakes  him  ?  Find  also  how  long  B  should 
start  after  A,  in  order  that  A,  when  overtaken,  may  have 
walked  G  miles. 

259.  Simplify  bVSa^  +  4aV^*  -  Vl25a«^*. 


MISCELLANEOUS   EXAMPLES.  XV 

260.  If  the  first  term  of  an  a.  p.  be  G,  and  the  sum  of  7  terms  105, 
find  the  common  difference,  and  shew  that  the  sum  of 
n  terms  :  sum  of  7i  -  3  terms ::  ?i  +  3  :  7i  -  3. 

261.  Which  is  the  greater  of  the  ratios 

a  +  2x:a  +  3x  and  a^  +  2ax  +  2x'^ : a"^  +  ^ax  +  ox'  1 

262.  Of  12  white  and  6  black  balls  how  many  different  collections 
can  be  made,  each  composed  of  4  white  and  2 blackballs? 

263.  (\)(x-ll)(x-2^)  =  ia^ix)(x-l) 

00  ^x-iij  +  z  =  7,ix  +  y-iz  =  l,y  +  iz-x  + 10^:0 

264.  A  market-woman  bought  eggs  at  two  a  penny,  and  as  many 
more  at  three  a  penny ;  and,  thinking  to  make  her  money 
again,  she  sold  them  at  five  for  twopence.  She  lost,  how- 
ever, 4.d  by  the  business  :  how  much  did  she  lay  out  ? 


265.  Shew  that   (x  +  ar')^  -  (y  +  y-^Y  =  (xy-xrY')  (xy''  -  ^'y\ 
and  exemplify  this  result  numerically  when  aj  =  i,  y  =  -  f . 

266.  Find  the  g.  c.  m.  of  4qjV  +  9«V  +  2ax''  -  2a^x-AaudzJx^ 

+  6ax^  -  ct^x  +  2. 

267.  Find  by  Evolution  Va  +  6x  to  five  terms,  and  square  the  result. 

268.  Simplify  3a-[Z^+  \2a-(h-x)]]  +i-f 


2x  +  r 


14  9 

269.  Find  the  sum  of  ,^ ;;  - + 


2ic  +  2     x+  2      2  (ic  +  3)      lx  +  2)(x  +  3)' 

270.  A  gamester  loses  J  of  his  money,  and  then  wins  10s ;  he 
loses  -J  of  this,  and  then  wins  £1,  when  he  leaves  off  as  he 
began.     What  had  he  at  first  ? 

271.  The  sum  of  n  terms  of  the  series  21  +  19  +  17  +  &c.  is  120 ; 
find  the  n^^  term  and  n, 

272.  Divide  100  into  two  parts  so  that  one  shall  be  a  multiple  of 
7  and  the  other  of  11. 

273.  Into  how  many  different  triangles  may  a  polygon  of  n  sides 
be  divided,  by  joining  its  angular  points  ? 

274.  Convert  85  and  257  to  the  quaternary  scale  ;  multiply  them 
in  that  scale,  and  reduce  the  result  back  to  the  denary. 

275.  (i)  ix  +  ^x-l  =  i  {Zx-i(x-l)} 

(ii)  ax  +  y  =  x  +  by=  ^(x  +  y)  +  1.      (iii)  Zx^y  =  144  =  4xy^ 

276.  A  and  B  can  reap  a  field  of  wheat  in  m  days.  B  and  C  in  ?i 
days,  and  A  can  do  p  times  as  much  as  C  in  the  same  time* 
in  what  time  would  the  three  reap  it  together  ? 


XVI  MISCELLANEOUS   EXAMPLES. 

277.  Find  the  value  o{  ax  -^ly  -  c  when 

mc  -  nh      .        Ic  -  na 

X  = y>  and  y  =  y. . 

7na  ^  lb  lb  -  ma 

278.  When  ^  =  4,  ic=-8,  2/  =  l,  shew  that 

a^x''  -^y^  ^  {a'^x^^-y^){a-'x^  -  a'^x^  y^  + 

279.  Reduce  to  its  simplest  form 

3a  V  +  5a 'x  -  12 


a-V-8^rV~12a-^a;  +  63* 

280.  Find  the  l.  c.  m.  of 

ax'^-lj  ax^  +  1,  (crx-iy,  (a'-'x  +  1)",  a-x^  - 1,  a-x*  +  1. 

4  1 

281.  Obtain  the  square  root  of  x^  -  Ax  +  ^x^  +  4. 

282.  Simplify  \jA0-^%jZ20+yn^,  and  8vJ-i  V12+4  V27-2VrV 

283.  Shew  that  the  sum  of  the  cubes  of  any  three  consecutive 
numbers  is  divisible  by  three  times  the  middle  number. 

284.  lia:b'.:c:d^  shew  that  2a''  -  W  -.  2c^  -  ZcP ::  a""  +  P:c^  +  d\ 

285.  Two  thirds  of  a  certain  number  of  poor  persons  received 
Is  GcZ  each,  and  the  rest  2s  (jd  each:  the  whole  sum  spent 
being  £2  155,  how  many  poor  persons  were  there  ? 

2SG.  The  No.  of  Comb""  of  n  letters  taken  5  and  5  together,  in  all 
of  which  a^  &,  and  c  occur,  is  21 :  find  the  No.  of  Comb"  of 
them  taken  G  and  G  together,  in  all  of  which  a,  5,  c.  d,  occur. 

1  2    _  '  3 

ic+3     iC  +  G     a;  +  9 


287.  (i)  VfcB  +  (1  -  xy  =  1  -  ir.         (ii) 


(iii)  a;'  +  icy  +  2/'  =  37,  x  +  y  =  7, 
288.  A  certain  number  of  sovereigns,  shillings,  and  sixpences 
amount  together  to  £8  Os  G^,  and  the  amount  of  the  shil- 
lings is  a  guinea  less  than  that  of  the  sovereigns  and  1^ 
guinea  more  than  that  of  the  sixpences :  how  many  were 
there  of  each  ? 


289.  What   is   the  difference  of  a  (b  +  cy  +  h  (a+cy+c(a+iy  and 
(a+h)  (a-c)  (b-c)  +  (a-b)  (a-c)  (b  +  c)-(a-b)  (a  +  c)(b-c)7 

290.  Prove  the  preceding  result  when  a  =  -  -^j  2)  =  1,  c  =  -  J. 

291.  Multiply  1+ia'^ x-^ia'x^hjl-^ ar^ X  +  ^a"^  x"^-^  a  V. 

292.  Obtain  the  coefficient  of ;?,« in  (1  -  2a;  +  Sa-'-'  -  Ax^  +  &c.y, 

293.  Extract  the  square  roots  of  7-^,  .064,  and  31  -  10  V6. 


MISCELLANEOUS    EXAMPLES.  XVll 

294.  Simplify 

295.  Given  two  numbers  such  that  the  difFerence  of  their  squares 
is  double  of  their  sum,  shew  that  their  product  will  be  less 
than  the  square  of  the  greater  by  the  double  of  it. 

20G.  Sura  to  n  terms  — ^r  +  -^  +  -^  +  &c.  and  -  +  1  +  -  +  <i;c. 
TT      n^      rr  n  a 

297.  Required  two  numbers  whose  sum  shall  be  triple  of  their 
diflercnce,  and  less  than  50  by  the  greater  of  the  two. 

298.  The  No.  of  Comb'''  of  n  +  1  things,  taken  n-\  together,  is 
3G  :  find  the  number  of  Permutations  of  n  things. 

299.  (i)  {a-vx){h^-x)-a{l)+c)  =  a''ch'+  x"" 

(ii)  ^/x  +  ^a-  X  =  2  \^Jx-^/a-x}  ♦ 

(iii)  2x^  +  S7f  =  5  =  -5 (2x  +  Zy) 

300.  A  can  do  a  piece  of  work  in  two  hours  which  B  can  do  in 
4  hours,  and -S  and  6^  together  in  1^  hour:  in  w^hat  time 
could  they  do  it,  working  all  three  together  ? 


301.  Divide 

12aj-20a;*  y~'^'  +  27a;'^  y'^  -  18x*  y-^  +  4?/"^  by  4a;^  -  4x^  y'^  +  y'i. 

302.  Fmd  the  value  of pr-  + —,  when  x r. 

x-2a     X-21P  a+  b 

303.  Extract  the  square  roots  of 

18945044881  and  (x  ^x'y-4(x-x '), 

304.  Find  the  g.  c.  m.  of  (b-  c)x'^  ^2  {ah  -ac)  x  +  arh  -  a^c  and 
{ah-  ac  +  h^  ~ic)x  +  {a'^c  +  ah"^  -  crh  -  ale), 

305.  Simplify 

V128  -  2  V50  + V72  -  Vl8;  and  (5  V5  -  7  ^2)-^{^b  -  2  V2/. 

306.  Find  the  sum  of   ^^^^         2/ -  ^       <c^ -y' 


2x  -  2y     2x  +  2y     x'  +  y'* 

307.  When  are  the  hour  and  minute  hands  of  a  watch  first  to- 
gether after  12  o'clock  ? 

308.  Expand  (3a"J  -  2a  ^x^^  to  five  terms. 

309.  Sum  T 2  +  i  +  ^  +  <^c.  to  8  and  to  Zn  terms ;  and  insert  four 
II.  means  between  f  and  f . 

310.  The  No.  of  Comb"  of   io  letters,  r  -  1  together :  No.  of 
Comb"  of  them,  r  +  1  together::  21 :  10:  find  r. 


XVlll  MISCELLANEOUS   EXAMPLES. 

311.  (i)  mx'  -  2.7x  =  30 

(ii)  (x  +  a)(y-  I)  +x  =  (x  -  a)  (y  +  h)-c) 
(x  +  h)  (y  -  a)  =  (a;  +  a)  (y-l)       ) 
(iii)  X  +  y  =  ax  +ly  =  ax^  -  by^ 

312.  Supposing  in  (300)  A  to  begin  by  himself,  how  long  af^cf 
must  ^  and  6^  begin  to  help  him,  so  that,  when  the  work  is 
finished,  J.  may  have  done  upon  the  whole  twice  as  much  as  <7? 

313.  Obtain  the  product  of  ^Ja  +  s/ax  +  ^.p, 

(i)  by  ^a  -  Vox  +  V-^j  (")  ^J  V-^  ~  ^^^  -  V^- 

314.  Find  the  value  of  -— =zL -sz:::^  whenaj=(i)||a,  (ii)  r-  yj 

va+x-  wa-x  "  i<o- 

315.  Write  down  the  quotient  of  16aV  -yhy  2a'^  x'^  +  2/  . 

316.  Extract  the  square  root  of  J  -  >>/5,  and  of 

25?-  -  ^^-xy'  +  j%x-Y  -  -'^x-'y  +  ^\  x''y-\ 

317.  Expand^/ and  a/-^—.  each  to  five  terms. 

^       y  a-x  y  a  +x' 

318.  Multiply  together 

1  +  2V2,  4- V3,  V2  +  V3,  4  +  V3,  2V2-1,  V^- V^. 

319.  Find  the  n*'^  term  and  the  sum  of  ?i  terms  of  the  a.  p. 

a  —  n     a-  2n     a  -  ^n 


■  +  ' + 


+  &c. 


n  n  n 

320.  If  the  sum  ordifTerencc  of  two  numbers  be  1,  shew  tUiitthe 
difference  of  their  squares  is  the  difference  or  sum  of  the 
numbers  respectively.  ^ 

321.  A  servant  agreed  to  live  with  his  master  for  £8  a  year  and 
a  livery,  but  was  turned  away  at  the  end  of  7  monUis,  and 
received  only  £2  13s  4fZ  and  his  livery :  what  was  it  worth? 

322.  How  many  different  sums  might  be  made  of  a  sovereign, 
half-sovereign,  crown,  half-crown,  shilling,  and  sixpence  ? 
and  what  would  be  the  value  of  them  all  ? 

...Ix^a     x-h      ^ax  +  (a  -  5)' 

OZO.    (1)     = =    =  T 

^^       b  a  ab 

,..^x  +  14-       a;  +  12        1      .....  ^  ^ 

^")^T2      2(":^TT9)==2     ("'^  ax-€y  =  0  =  ayi  hx-cxy 

E24.  Two  girls  carried  between  them  25  eggs  to  mark«;t :  they 
sold  at  different  prices,  but  each  received  the  sam^  amount 
upon  the  whole :  the  first  would  have  sold  them  c.ll  for  1a- 
the  second  for  13c?:  how  many  did  they  each  sell  ? 


MISCELLANEOUS   EXAMPLES.  XIX 

325.  Write  down  the  square  of  1  -  -|a;  +  ix^,  and  square  the  result. 
32G.  Divide  -2x''ij-^+17x'ij-'-6x'''2-LxY  hy-x^i/-^+7x'^y~'  +  SxY' 

327.  Find  ^7,  and  thence  V?,  V^Ij  V^i  "^  V^ij  2  -*-  (4  -  V"^). 

328.  Find  the  value  of  ^ ^ —  +  77— j — ^r — ,  vrhen  x  =  i  (a  -i  I). 

2na-2nx     2nb-2nx^ 

329.  Simplify  VJi^  %^  and  |^-^^ V^" 

ooO.  Find  the  sum  of  ^  .  ^  -  -  -  ^-^^-^^  .  ^,-^  -  ^^.-^y),. 

331.  If  w=^+^  +  rj  where  p  is  constant,  ^  oc  xy^  and  7'oc  ^1*2/"^  and 
when  a;  =  2/  =  1,  -w  =  0,  when  a;  =  y  =  2,  w  =  6,  and  when  aj  =  0, 
-i^  =  1,  find  u  in  terms  of  x  and  2/. 

332.  Shew  b}^  the  Bin.  Theorem  that  V3  =  l+f  -  -J+||-  if  J  +  &c. 

333.  In  how  many  wa}- s  could  I  distribute  exactly  555  among  the 
poor  of  a  parish,  by  giving  Is  Q>d  to  some  and  2s  (Sd  to  others  ? 

334.  How  many  words  can  be  formed  of  4  consonants  and  2 
vowels,  in  a  language  of  24  letters,  of  v/hich  5  are  vowels  ? 

335.  (i)^(..i]=i.-^f  .^(1.1 

a  -  c\       xj  (a  —  c)x     a  -  c\       x^ 

(ii)  4a;  -  5^/  +  mz  =  7x—  Ihj  +  nz  =  x  +  y  +  pz-  (^ 
336.  A  boat's  crew  rowed  3^  miles  down  a  river  and  up  again  in 
100' :  supposing  the  stream  to  have  a  current  of  2  miles  an 
hour,  find  at  what  rate  they  would  row  in  still  water  ? 


^  2ac       ^    , ,,        ,        ^Va  +  dx  +  Va  -  bx 

667.  li  x  =  ^r~z ~,  find  the  value  of 

^G+^)  ^a  +  bx-^a-lx 

QOQ    T>   1       *    -I-    1         -i          Sax^  -  2a^x'' -  ah 
3oO.  Reduce  to  its  lowest  terms 

Oco^x'  —  a^x  —  1 

330.  Find  the  coefficient  of  a;°  in  (I  +  ^a;  +  ^x^  +  |a;=  +  &c.)' 

340.  Fmd  the  sum  of 


a^^l)  -  1/      (a  +  1)1)     a-"  -  h^' 

341.  Simplify  a'bc  Var^c  -  ¥c  V^f^W^  +  a'l'c''  V2-iZar'h-''c-\ 

342.  Obtain  the  cube  roots  of  51.0G4811,  and  1  -Ga;  +  21a;'-  44a;' 
+  63a;*  -  54  a;^  +  27a;^ 

343.  The  prime  cost  of  38  gallons  of  wine  is  £25.  and  8  gallons  are 
lost  by  leakage:  at  what  price  per  gallon  should  the  re- 
mainder be  sold,  to  gain  10  per  cent,  upon  the  outlay  ? 


XX  MISCELLANEOUS   EXAMPLES. 

344.  Ua:'b'.:c:d,  shew  that 

5  3^ _5 

345.  Expand  {2fl-3  vax'f^  and  {Sa-2va^Xf  -,  each  to  live  terms. 

346.  From  a  company  of  50  men,  5  are  draughted  off  every  night 
on  guard :  on  how  many  different  niglits  can  a  different 
selection  be  made  ?  and  on  how  many  of  these  will  two  given 
soldiers  be  found  upon  guard  ? 

347.  (i)  ^^^^^^  =  ax.V  ('">  f 2'  =  '  f'  ^  "/I i 

^  ^       a  +  X  Ixy  -  c  {ax  -  by)  ) 

(ii)  5.r  -  l\y^+\Zz^  =  22,  ^x+Gy^+C^zi  =  31,  a;  -  y^'  +  s^  =  2. 

348.  A  person,  having  to  walk  10  mileSj  finds  that,  by  increasing 
his  speed  half  a  mile  an  hour,  he  might  reach  his  journey's 
end  IGf  minutes  sooner  than  he  otherwise  would  :  what  time 
will  he  take,  if  he  only  begin  to  quicken  his  pace  halfway  ? 


349.  Divide  (.r'» -I)  a^-(x^  +  x^-2)a''  +  (4a;'  +  Zx  +  2)a-Z  (x+1) 
by  (x  - 1)  a^  -(x-1)  a  +  3. 

350.  Multiply  l/a"^  +  ^(a^c)^  by  ^a^-\/(a^c)^.  ' 

351.  Ifa;=  Vi-4^+  V  (i^'"  -  2?!?')  h  ^^n^  the  value  Grx'^rx^+^\q\ 

352.  Extract  the  square  root  of  ^x^-^x^y^ ■h-\Y-^^y-l^'^y^ +^%xy\ 

353.  Addtogethor-J4Ll^^^^  and  -,Kip^)^l_; 

^  a;^  -  -^  (1  +  ^5)  aj  +  1  aj^  -  i  (1  -  V^)  a;  +  1 

354.  Find  the  sum  to  n  terms  and  ad  ivf,  of  the  g.p.,  whose  first 
two  terms  are  the  a.  and  11.  means  between  1  and  2. 

355.  "What  is  the  least  number  which  is  divisible  by  7  and  11 
with  remainders  G  and  10  respectively? 

35G.  A  privateer,  running  at  the  rate  of  10  miles  an  hour,  dis- 
covers a  ship  18  miles  off,  making  away  at  the  rate  of  8  miles 
an  hour :  how  long  will  the  chase  last  ? 

357.  Expand  -J  2^^  -  3  yax]  ^  and  { Za -  2  va-a?[  -,  each  to  five  terms. 

358.  In  what  scale  will  the  common  number  803  be  expressed  by 
30203  ?  What  are  the  greatest  and  least  common  numbers 
that  can  be  expressed  with  five  digits  in  it? 

359.  (i)--i^''-.^-  =  .  (ii)^    i^=<, 

h  +  x      0 -X  ^     b  +  X      b -X 

X      y  x-a     y-b 

(m)  --  +  -  ~l  -  — —  + 

^    "^  a      b  b  a 


MISCELLANEOUS   EXAMPLES.  XXI 

360.  -4,  B^  C  reaped  a  field  together  in  a  certain  time :  A  could 
have  done  it  alone  in  9  J  hrs  more,  B  in  half  the  time  that  A 
could,  and  C  in  an  hour  less  than  B.  What  time  did  it  take 
them? 


301.  Divide  sJx'Y  -  z  -Jx'y''  -  f aj  VV  +  f ^V  Vjj" V' 

by  \/xi/  -  "I  -Jx^y^. 
3G2.  The  edges  of  three  cubes  are  a,  h,  a  +  h;  shew  that  tho 

greatest :  difference  between  it  and  the  sum  of  the  others 

363.  Extract  the  square  root  of  a;  +  1  -2^0?  (1  +  ^x)  +  3  ^x. 

364.  Simplify  ^72  -  3  VJ  and  V2^^  -  V2ax^-4ax-i-  2a, 

365.  If  oj  =  ^  (V3  +  1),  find  the  value  of  4  (x^  -  2x^)  +  2x  +  3. 

366.  A^s  money  with  ^  of  i?'s  would  be  ^  as  much  again  as 
before  ;  and  if  2s  be  taken  from  JL's  present  sum  and  added 
to  i?'s,  the  latter  amount  will  be  I  of  the  former.  What  had 
they  each  at  first  1 

367.  Find  the  value  of  Va  +6x,  and  square  the  result. 

368.  If  the  difference  of  two  fractions  be  vi?i  ^,  shew  that  m  times 
their  sum  =  n  times  the  difference  of  their  squares. 

369.  The  first  term  of  an  a.  r.  is  71^  -n  +  1,  the  common  differ- 
ence 2 ;  find  the  sum  of  n  terms,  and  thence  shew  that 
1  -  IV  3  +  5  -  2^  7  +  9  +  11  =  3^  &c. 

370.  Find  the  area  of  a  court  250  ft  long  by  200  ft  broad,  (i)  by 
the  senary,  (ii)  by  the  duodenary  scale. 

371.  (i)  — + = (ii)  7ix  +  -  =  na  ^ha^ 

^  ao  -  ax      be  -  ox      ac  -  ax  x 

B       (iii)  a;^  +  7/^  =  2a'^,  x  +  y-.x-y.-m-.n 

^72.  A  cistern  has  three  pipes  A^  B,  and  C:  by  A  and  B  together 
it  can  be  filled  in  36',  and  emptied  by  C  in  45',  whereas,  if  J. 
and  C  were  opened  together,  it  would  be  emptied  in  1^  hr : 
in  what  time  would  it  be  filled,  by  A,  by  B,  or  by  all  together '? 


Ill  n 

373.  Find + + ,  when  s  =  —  (171-11+  p), 

mn  -  mz     np  —  nz      mz  —  mp  m 

374.  Multiply  7?iaa;^  +  {m-l)a''x'^  +  (m-2)  a^x  ^  by  a'^  \Jx*Ajx, 

375.  Extract  the  square  root  of  1  +  m^  +  2  (1  -  ni^)  -^m  +  Sw-?7i'* 


XXll  MISCELLANEOUS   EXAMPLES. 


37G.  Simplify  -^l-  ±  l/.-^,  .  ^  .  ,__. 

'   x-y      Y    {x-yy     x-y     x^\lxy 

377.  Find  a  number  of  two  digits  such  that  its  quotient  by  their 
Bum  exceeds  the  first  digit  by  1,  and  equals  the  other. 

378.  IIow  many  terms  of  the  scries -7 -5-3  -  &c.  amount  to 
9200?  and  how  many  of  G  +  4  +  2J  +  &c.  amount  to  14|? 

379.  A  certain  number  of  men  mowed  4  acres  of  grass  in  3  liours, 
and  a  certain  number  of  others  mow  8  acres  in  5  hours :  how 
long  would  they  be  in  mowing  11  acres,  all  working  together  ? 

380.  If  «,  J,  Cj  d  are  in  c.  p.,  shew  that 

(r/.  +  5  +  c  +  (Tf  =  (6?  +  Vf  +  (c  +  dy  +  2(Z;  +  cf, 

381.  The  No.  of  Var°"of  n  things,  r  together:  the  No.,  r-1  to- 
gether-10  :  1,  and  the  corresponding  Nos.  of  Comb"'  arc  as 
5:3;  find  n  and  r. 

382.  A  person  makes  20  lbs.  of  tea  at  4^  OtZ,  by  mixing  three 
kinds  at  3^  6^,  45  CcZ,  and  ^os  :  how  can  this  be  done  ? 

383.  (i)x(^.i^5)_Ll^^^^_.^i_i5^_|(l_.3^.)i 

(n)  x  +  a  +  h  +  c  = (in)     -      +  [f]  =2} 

a+  b  +  c  +  X  ^        \aj        \h  J         \ 

ay  +  hx  =  0  J 

384.  A  trader  maintained  himself  for  3  years  at  an  expense  of  £50 
a  year,  and  in  each  of  these  years  increased  tliat  part  of  his 
stock  which  was  not  so  expended  by  -J-  thereof:  at  the  end 
of  3  years  his  original  stock  was  doubled :  find  it. 

385.  Divide  (C«=-7(jJ  + 260  a''  +  (5a'-3a=Z>-5a2*H36V+  (fi'-i^')'^ 
by  {2a  -  h)  x  +  cC  -  h\ 

3SG.  Find  the  l.  c.  m.  of 

X*  -(p^  +  l)x'  +  p^  and  x*  -  (p  +  ly  ic'  +  2  (^  +  1)  px  -p*, 

387.  Obtain  the  values  of  (i)  x  -  ^fxy  +  y,  and  (ii)  of  x^  ^  xy  ^  %f^ 
when  X  =  y^g  (4J  +  V"t);  V  =  iV  (H  -  V^t)- 

388.  Simphfy  (a-^l)\  — i-^,  +  --!-„  ^  +  2  ^  -^ LI 

l(x+  ay      (x  +  by)        (x  +  a     x  +  b\ 

389.  Obtain  the  square  roots  of 

1V2       -2v/2  a'c         ^      ^  If 

2  +  a       +  rt        and  —  -^  cj  -  2ac  a/  ^ 

390.  The  n^  term  of  an  a.  p.  is  ^n  -  J  :  find  the  sum  of  n  terms. 


MISCELLANEOUS   EXAMPLES.  XXlll 

391.  The  diagonal  of  a  cube  is  a  foot  longer  than  each  of  the  sides : 
find  the  solid  content. 

392.  Find  the  first  time  after  noon  when  the  hour  and  minute 
hands  of  a  watch  point  exactly  in  opposite  directions. 

393.  In  how  many  ways  ma)^  £10  be  paid  in  crowns,  scvenshilling 
pieces,  and  moidores  (27s)  thirty  coins  being  used  ? 

394.  Out  of  5  white,  7  red,  and  8  black  balls,  how  many  different 
Bets  of  G  balls  could  be  drawn,  (i)  two  of  each  colour,  (ii)  one 
white,  two  red,  three  black,  (iii)  three  red,  three  black  ? 

395.  (i)  X  +  Vaj'  -  2ax  +  h""  =  a  +  h 


(u) = (in)  y  1  +  ~  +  /4/  1  — -  =  1| 

X  ^  a     X-  c     X  +  x-c  r  cr       f         a^         ^ 

S9G.  Two  vessels,  A  and  B^  contain  each  a  mixture  of  water  and 
wine,  A  in  the  ratio  of  2  :  3,  ^  in  that  of  3  :  7.  What  quan- 
tity must  be  taken  from  each,  to  form  a  mixture  which  shall 
consist  of  5  gallons  of  water  and  11  of  wine  ? 

397.  Shew  that  {ay  -Ix)'  +  {ex  -  azy  +  (Iz  -  ct/Y 

=  {a^  +  &^  +  c^)  {x"  +  y^  +  s^)  —  {ax  +  hy  +  czf, 

398.  Find  the  g. c.  m.  of  Zx^  +  (4a  -2V)x-  ^al  +  a""  and 

x^  +  {2a  -d)x^-  {2ah  -a'')x-  a'5. 

399.  From-i  {x^  +  Zx'^)  {x^-^x"^)  take  i  {x^  +  2x'^)  {x^  -  3a;"^), 
and  multiply  the  result  by  6  (1  -a;^)'\ 

400.  Extract  the  square  root  oix'^  ■^2x'^  +  ?>x^  -2x  ^  ^  x  ^  -  1. 

401.  Multiply  together 

n+l  m-\  n-1 

''yjaTbyf'^\  ^{a  +  hy^^  ^{a  +  ly^^  y{a  +  Z))"2  . 

402.  Simplify 

\l  +  x       4:X         Sx       l-x}   ^  jl+g;^       Ax''       l-x^l 
(1^  '^l  +  x''^  1+x'      U'x)   '    il^x^'^Ui^T^^S 

403.  Sum  {a  +  ic)^  +  {a^  +  x^)  +  {a  -xy  +  &c.  to  5  and  to  n  terms. 

404.  Find  two  numbers  such  that  their  sum,  product,  and  differ- 
ence of  their  squares  may  be  equal. 

405.  Apply  the  Bin.  Theor.  to  find  (l.Ol)"^  to  nine  places. 

400.  Find  the  least  integer  which,  when  divided  by  7,  8,  9,  re- 
spectively, shall  leave  remainders  5,  7,  8. 
407.  (i)  a;  +  3  =  V2(a;-r3)  +  4         (ii)  ahx'  -  {a  +  5)  ex  +  c^  =  0 

«(iy*(i)"='.M=«- 


XXlv  MISCELLANEOUS   EXAMPLES. 

408*  A  person  bought  38  sheep  for  £57 ;  but,  having  lost  a  cer- 
tain number,  n^  of  them,  he  sold  the  remainder  for  n  shil- 
lings a  head  more  than  they  cost  him,  and  so  gained  upon 
the  whole  IGs:  how  many  sheep  did  he  lose? 


400.  Shew  that  (a^  +  5'  -  1)''  +(a"  +  V'  -1)^+2  (aa'  +  Uy 
=  (a"  +  a""  -  ly  +  Q)^  +  &"  -1)^  +  2  {ah  +  a'h'y. 

410.  Find  the  g.  c.  m.  of  xy  +  2x^  -  Zy^  +  Ayz  ■¥  xz-z^  and 

2^'  -  ^xz  -  ^xy  +  4s'  -  ^yz  -  127f. 

411.  Find  tho  foiirth  term  of  (^2  +  ^3)",  correct  to  four  places. 

412.  Obtain  the  square  root  of  l  +  x-^\/x  (1+  ^Jx)  +  ^Jx(2+  j%^x). 

413.  If  the  r^^  term  of  a  series  be  a?^  ^  -7*,  shew  that  the  sum  of 

.u       ,    »v  -,    .1     ^         N»vi     m^+mn  +  n'* 

the  m*^  and  n^^  terms  exceeds  the  (?7i-f??r''  by ; r  a. 

mn  {m  +  n) 

414.  If  X-'  =  (a-c)  (h- c),  y'^  (a -h)(h- c),  z'  =  (a-h)(a-  c), 
find  the  values  of  a;  —  y  +  ^  ^^^  ^^^^  ~  ^^Z/  +  ^^^* 

415.  If  P,  §,  i?,  be  the  jp"*,  q^^  and  r*^  terms  of  any  h.  p.,  shew 
that  (p-q)  FQ  +  (q-r)  QE  +  (r-^?)  EP  =  0. 

41C.  Two  parcels  of  cotton,  weighing  0  lbs  and  16  lbs,  cost  lis  6d 
and  £1  Os  Ad  respectively^,  and  (he  charge  for  carriage  was 
proportional  to  the  square  root  of  the  weight :  how  much 
per  lb.  was  paid  for  the  purchase  of  the  cotton? 

417.  If  a  :  Z> ::  &  :  c,  shcw  that  a  +  h-.l)  +  c::a^  (h-c)  :h^  (a^h), 

418.  Find  tho  least  number  which  being  divided  by  2,  3,  5,  shall 
leave  remainders  1,  2,  3. 

419.  (i)  (x-1)  +  2  (x-2)  +  3  (a;-3)  +  &c.  to  six  terms  =  1         4 

....  2x(a-'X)      ,  .....  X     y     ^      X     z 

420.  A  square  court-yard  has  a  rectangular  walk  around  it ;  tho 
side  of  the  court  wants  2  yds  of  being  six  times  the  breadth  of 
the  w^alk,  and  the  no.  of  sq.  yds.  in  the  walk  exceeds  by  92 
the  no.  of  yds  in  the  periphery  of  the  court :  find  its  area. 


ANSWERS  TO  THE  EXAMPLES. 


1. 1. 

48. 

c. 

-1. 

2.  1. 

11. 

6. 

-64. 

3.  1. 

25. 

6. 

22. 

4.  1. 

46. 

6. 

135. 

2.  12. 
7.  -178. 

3.  -8. 
8.  150. 

2.  1. 

7.  16. 

3.  0. 
8.  264. 

2.-15. 

7.  7. 

3.  12. 
8.  13. 

2.  24. 
7.  8. 

3.  35. 
8.  120. 

4.  1.  5.  106. 

9.  450.      10.  192. 


4.  94. 
9.  5. 

4.  6. 
9.  15. 

4.  10. 

9.  384. 


5.  89. 
10.  3. 

5.  21. 
10.  4. 

5.  7200. 
10.  4. 


5.  1.  15a  +  Zh-Gc+  ed, 
3.  23a^-26ab+Uh\ 
5.  5aj'  +  50a;V  -  Uxi/  +  4yK 
7.  -  9a;*  +  2ax^  -  ZWx  +  16^^ 
I     9.  Cx^  +  4f  +  2"  -  24a;y;?. 


2.  14a;  -  9y  +  10«  - 12. 
4.  Ghij  -  7cz, 
6.  2x'  +  2if  +  22'. 
8.  a*  +  b^  -\-  c*  +6a6(?. 
10.  ic*  +  y*  +  2* 


I 


6.  1.  «  -  35  +  3c.        2.  -  2a;'  -  7a;y  +  Zy\       3.  4aa;  -  95y  +  2cz. 
4.  5a;'  -  5a;  +  5.  5.  7a'  -  3a  +  4&'  -  7ah  +  2c'  -  CJc. 

6.  -  a;'  -  6a;'y  -  2?/'  +  6  -  3a5'  -  4y' 

7.  3a;' +  13a;y  -  ^f  ^  16x2  -  13y2.  8.  x^  +  xij  +  y\ 
9.  3a*  -  4a'b  -  4a&»  +  25*.                             10.  0. 


7.  1.  4a -4a;.  2.  4a»  -  4a'c. 

L4.  2aa;'  +  25y'  +  2c2'.     5.  a'  -  36'  +  3c'. 
7.  0.        8.  "Zx-y+iz.      9.  8a; -8. 


3.  a;'-3y'-32». 
6.  2a5  +  45'.  . 
10.  -  4c  +  4d!. 


O) 


ANSWERS   TO   TIIE   EXAMPLES. 

.  1.  (2a-h)-(Sc-^d)-(2e-2f),(2a-h-Zc)  +  (U-'2e+Zf), 

2.  -(&  +  Zc)+(4cd-2e)  +  (3/+  a%-(b  +  3c -4^ -(26 -3/- a). 

3.  -  (3c  -  M)  -  (2e  -  3/)  +  (2a  -  h\  -  (3c  -  4^  +  2e)  +  (3/+  2a-hy 

4.  (U-2e)  +  (3/  +  2a)  - (Z>  +  3c),  (4d-2€  +  3/)  +  (2a  -  5- 3c) 

5.  _  (2c - 3/)  +  (2a -b)-  (3c - 4fZ),  - (2c -  3/- 2a)  -(b  +  Zc-id) 
G.  (3/  +  2a)  -  (6  +  3c)  +  (U  -  2c),  (3/  +  2<x  -  Z>)  -  (3c - 4d;  +  2c). 
7.  {2a-(h  +  Sc)\  +  {id-(2e-^f)\.  8. -|6  +  (3c-4fZ)}-{2c-(3/+a)}. 
9.  -{3c-(4^-2c)}  +  j3/+(2a-&)}.10.  { 4tZ-(2c-3/) |  +  { 2a-(&  +  3c) 

11.  -j2c-(3/+2«)}-lZ>+(3c-4cr)}.  12.   {Sf+(2a-b)]-{Sc-(4d'-2€)\ 


1,  (a^b  +  c)  x^ - (b -  c  +  d)  x^ ^  (c  +  d  +  e)  X,         2.  2(ax -  by). 
3.  (a  +  b)  x^-  (a  -  55)  a??/  +  (^^  -  c)  i/'*.      4.  2  (aaj  +  cy),  2b  (x  +  y. 

5.  ^(a-bb)^  +  (2a  +  Zb  +  c)y^  (a-4b-c)  x  ^-  (a~Zb- 2c)y, 

(b-c)x  -\-  (3<x  -  c)  y. 

6.  (5a  -  5)  a;  -  (2a  -  35  -  5c)  y,  -  (a  +  c)  ic  +  (a  -  5  +  2c)  y, 

(4a  -  5  -  c)  a;  -  (a  -  25  -  7c)  y. 

7.  (2a  +  45  +  c)a;-(a-56-3c)2/,  -(4a-55)ic  +  (2a  +  5)y, 

-(2a-95-c)a5+  (a  +  (Sb  +  3c)y. 

8.  (a  +  45) ic  +  (45  +  5c) y,  -(3a-55-c)a;  +  (a  +  25-2c)y, 

(2a  -  95  -  c)  aj  +  (a  +  05  +  3c)  y. 


10.  1.  abx^ij*,  -  mnx^^  2a'^cx^y,  a¥c^,  a^bc^^  -  x^ij^, 

2.  x^-x^y  +  icy^,  -a^a;  +  aV-aa;',  -a5a;'  +  a^bx^-ab'^x^ 

x*y  -  Sa;^^^''  +  Zx'^y^  -  a^y*. 

3.  2a''  +  7a5  +  35»,  2ac-5c-6acZ  +  35^. 

4.  (jx''  +  13a;y  +  6?/^  6a^5'-a5«-125^ 

5.  x""  +  Gaj''  +  7a;- 6,  x^-^yx''  +  11a;- 6. 

6.  a*  +  a'-2a=  + 3a-l,  a*-a^-8a^ +a  +  1.      • 

7.  81a;*-2/*.  8.  a=^  +  325^  9.  a'*-4a''a;+3a\ 
10.  27a»+5^  +  8-18a5.     11.  a;^-?/»+2»+3a'2/^.     12.  a«-l. 

13.  a«  -  85»  ~  27c*  -  18a5c.  14.  a«  +  2a»5*  +  5». 


15.  a;'*  -  (a  +  c)  a;'' +  (ac  +  5)  a;  -  5c ;  a;^-(a^-5  +  c)a;''+a(5 +c)a;-5c. 

16.  l-(a-l)a;-(a-5  +  l)a;^  +  (a  +  5-c)a;*-(5  +  c)a;*  +  ca'*. 

17.  a^  -  amx  -  2m''aj'*  +  2>mnx^  -  n^a;* ; 

a''  +  a(m+ 2n)  x-  \  a  (m + /i)  - 2mn ]  x^-  (in?  +  2n^)  x"^  +  mnx*, 

18.  a'x*  -  a*-*  (5  -  c  +  cZ)  x'^y  -  (abc  -  abd  +  acd)  xy'^  +  bcdy^. 

19.  4x*  +  6  (w-7i)  a;'-  (4m^  +  9wi/i  +  4ii^)  %^  +  Cm7i  (w-;?)  a;+ 4w'/i*. 

20.  a;*-(2a^+25»  +  a5)a;='+  (aH  a^5+  a'b''^aV^V)%'-(a^l)a'b\ 

(2) 


ANSWERS   TO   THE   EXAMPLES. 


11.   1. 

2. 

4. 

5. 

7. 

9. 
10. 
11. 
12. 
13. 

14. 


a'''-2ax+x\  l  +  4ic'  +  4x*,  4a*  +  12a*  +  9,  9a;' ^24ji/  +  16y\ 

9  +  12aj  +  4x\  ix^  -12xy  +  Oy\  a*  -  Wx  +  9a'a;-, 

V'x''  -  2hcx^y  +  cV^/'.  3.  4a'  -  1,  9<zV  -  h\  a;*  -  L 

x"  +  4a;  +  3,  a;*  +  3a;'  - 4,  a^l^  -ah -6,  4a V  -  Sahx  +  3&'. 

a;*-  5aV  +  4aS  6.  7?iV  -  IZmhi^xY  +  36^i*2/* 

4a;'.  8.  x^  +  4^/^  4a*  -  5a'6'  +  Z^*. 

a'  +  2ah  +  5'  -  c',  a'  -  &'  +  2ac  +  c',  a'  -  &'  -  2hc  -  c\ 

a'  -  2a&  +  &'  -  c',  -  a^  +  2a6  -  2^'  +  c',  -«'+?>'-  2hc  +  c». 

4a'  -  &'  +  G5c  -9c',  -  4a'  +  12ac  +  &'  -  9c'. 

4a'  -  &'  -  G5c  -  9c',  -  4a'  +  4a5  -  5'  +  9c'. 

a'  +  2ac  +  c'  -  Z;'  -  22>cZ  -  ^',  a'  +  2acZ  +  eZ'  -  5'  -  26c  -  c\ 

h^^2lc  +c'-a'-2a^-^'. 

a'+  2ad  +  ^'-46'+  12Z'c-9c',9c'+  6ccZ  +  <Z'-a'+  4a5-45', 

a'  +  6ac  +  9c'-46'  +  4?>^-cZ'. 


12.  ^.  &c',  5^2/',  -35Z?a;. 

2.  3a;y-2a;^+32/2,  -a-/>'  +  7a&c'-4c*,  ■ 


-3aa;+36y —  . 

a 


2??i         4??i' 
3?^  3/2,' 


5a  5      2Z»' 

2^'"^^2^"'«;^ 


n_      3^' 
3m'    26' 

4.  X  ■\-  5,  m'^'^m  +  3.  5.  3a-26,  3a;  +  2y. 

G.  2ab-U\  7.  a'-2a6+26',   2a;'i/'+2a;y+l. 

8.  'a;*-2a;V  +  4a;'y'-8a'i/V16i/*.  9.  l-2a;+3a;'-4.t;«+5a;*. 

10.  a;'+  2a;y  +  3?/',  m'-2m  +  3.  11.  a'+  2a'&  +  3aJ'+  46». 

12.  x'  +  2a;^  +  3a;'  +  2a;  +  1,  a*->2a'6  +  3a'6'-2a6'  +  h\ 


13.  1.  a;'-^a;*'+  ^.  2.  as'  +  6s--c. 

3.  y*-(ni^  1)  y^-(m- 71^  1)  y'  -  (w - 1)  y  +  1. 

4.  a  +  &  -  c  -  <Z.  5.  a  +  26  -  c. 

6.  a'  +  6'  +  c'  +  a6  -  ac  +  6c,  a'  +  6'  +  c'  +  a6  +  ac  -  be, 

7.  1  -  a;  +  2y  +  a;'  +  2xy  +  4^/',  1  +  a;  -  2y  +  a;'  +  2a;i/  +  4y'. 

8.  a;'  +  42/'  +  9^'  +  2.ry  +  Zxz-6yz.  9.  a;'  +  y'  +  «'  +  1 

^/p*  l3oa;* 

10.  a-ax+ax'-ax^  +  :;— — -  ,  1  +  5a;  +  15a;'  +  45a;'  + 


11.  1  +  2a;  +  3a;'  +  4x''  + 


1  +a;' 

5a;*  -  4a;^ 


l-3a;' 


l-2a;  +  a;" 


1  -  (a  +  6)  a;  +  (a  +  6)  6a;'  -  (a  +  6)  6'a;*  + 


12.  a^-pa' 
(3) 


(a  +  6)  Px* 
1  +  6^' 


qa-r. 


ANSWERS   TO   THE  EXAMPLES. 

14.  1.  a-'X,   a*+  a^x + a^x^  +  ax*  +  x\  a*  -  a*x + a*x^  -  a'*x*  +  ax*  -  x\ 
2.  Saj  +  l,  5a;-l,  2aj-3.  3.  3m7i-5,  4m'-7i'. 

4.  l-2aJ  +  4a;^  Ox''  +  Sx  +  l,  1  - 2a;  +  4x' - 8a;*. 

5.  ic'  +  Sx'y  +  Oa-y'  +  272/^  a*-2a*b  +  4a^6-  -  SaJ'  +  166*, 

6.  j^2_|a5+&'',  xY-^^y^^+^y^''-^^'        '^'  a+h  +  c^  a+h-c. 
8.  Or  +  2/)'-(^  +  2/)  0  +  ^'^  =  a;'  +  2xy  +  y'-xz-yz  +  0^ 

ic'  +  ic  (2/-0)  +  (y-s)^  =  x''  +  xy-xz  +  2/'  -  2^3  +  «^ 

15.  1.  (l-2a;)(l  +  2x),  {a-Zx)  (a  +  Zx\  (Zm-2n)  (3m  +  2n), 

ic^  (5a  -2)  (5a  +  2),  icV'  (4a;  -  5?/)  (4a;  +  57j). 

2.  (a;  +  2/)  (a;^  -  a;?/  +  y'),  (a;  -  y)  (x""  +  xy  +  y% 

(1 4-  xy)  (1  -  a;y  +  a;^2/'),  (^  -  1)  (^  +  1)  (P"  +  1)> 
xy  (ay-x^)  (ay  +  a;'),  2a¥c  (a -2c)  (a  +  2c). 

3.  x^  (ox  -  a)  (5a;  +  a),  a*  (a  -  ^¥)  (a  +  3i'), 

(2a;-  3)  (4a;^  -^  Cx  +  9),  (a  -  26)  (a'^  +  2ah  +  46^, 

4.  (X  +  2)  (a;*  -2a;'+  43;^^  -8a;  +  16),  x*  (a  +  3a;)  (a^-Zax  +  dx") 
(2x'  +  y')  (4a;«  -  2a;V'  +  y%  (ah'  +  c  =)  (a6^  -  c'^)  (a'6«  +  c*\ 
ahc  (a  +  cy. 

5.  (3a;-l)  (3a;+l)(9a;ni),  (a;-2)  (a;+2)  (a;V2a!+4)  (a;'~2a;+4). 
x^x-hy,  x''(x-ay(x  +  ay, 

6.  (4a; -5)  (2a;  +  1),  (a  +  36)  (a-&),  7  (x-y)(x  +  y), 

7.  (a;-2/)'  (a;  +  2/)',  (c  +  a- 6)  (c-a  +  6),  8a6. 

8.  (x  +  2/)',  ^/i  (m-n),  6&  (a- 6). 

9.  2(a;  +  2/)  (^x-y\  2  (x-y)  (^-x),  Ay(x  +  y), 
10.  (a  +  I)  (a''  +  a6  +  6^^),  (a-hy,  0. 

16.  1.  (a;  +  l)(a;  +  5),  (a;  +  4)  (a;  +  5),  (a; -  2)  (a; - 3),  (a;  -  3)  (a? - 5), 
(a;  +  1)  (a;  +  7),  (a;-l)  (a;-9). 
2.  (a;  +  3)  (a;-2),  (a;-3)  (x  +  2),  (a;-3)  (x  4- 1),  (a;  +  5)  (a;-3), 

(a;  +  8)(a;-l),  (a;"9)  (a;  +  1). 
8.  (2a;  +  3)  (2a;  +  1),  (4a;  +  1)  (a;  +  3),  (4a; -1)  (x  +  3), 
(2a;  -  3)  (2a;  +  1),  (3a;  -  2)  (x  +  2),  (3a;  +  4)  (2a;  - 1). 

4.  (4a;  4- 1)  (3a; -2),  2(6a;-l)  (a;-l),  (4a;  +  1)  (3a;  - 1), 
(x  +  4)(x-Z),  (3a;-5)(a;  +  l). 

5.  a-  (x-a)  (x-2a\  a  (a- 3a;)  (a  +  2x),  ah  (3a-  25)  (a  +  h\ 
(4a' -a;^  (3a' +  a;'). 

6.  xy  (2x  ^y)(x^  2y\  Sy'  (Zx + 2y)  (x  -  y\  a'  (Zax  - 1)  (2aa;+ U 
a;'  (26  -  3a;)  (36  -f  x), 

(4) 


ANSWERS   TO  THE   EXAMPLES, 


XV.  1.  5. 

2.  2. 

3.  3. 

4.  t-. 

5.  ^i. 

6.  '^-t 
m-n 

7.  2. 

8.  1. 

9.  4. 

10.  -1^. 

11.  -4. 

12.  |. 

13.  -f. 

14.  ^l 

n 

4.  5. 

18.  1.  42. 

2.  12. 

3.  12. 

5.  7. 

G.  4. 

7.  5. 

8.  h 

9.  7. 

10.  yV  (25a  - 

- 18^). 

11.  7. 

12.  -8. 

19.  1.  4.        2. 

2.        3. 

18.      4.  8, 

5.  -a. 

6.  6. 

7.  4.        8. 

l-a.     9. 

7.      10.  a-w. 

11.  10. 
4.  7.9. 

12.  2(a  +  c). 

20.  1.  12. 

2.  9. 

3.  120. 

5.  35,  13. 

6.  513,  46G 

i.      7.  15. 

8.  31,  : 

18. 

9.  15. 

10.  90,  60. 

11.  24  ft. 

12.  16. 

13.  37, : 

50,20.   14.  20. 

15.  41. 

16.  £5. 

17.  88. 

18.  85^, 

35s. 

19.  £36, 

£12,  £16. 

20.  5. 

21.  £45,  £57,  £63 

,  £65. 

22.  15,  5. 

23.  9S§  miles  from  L, 

lOf  h. 

24.  22,  7,  12  gals. 

25.  1  h  20'  i 

from  B's  s 

tartiiig,  (j\  milej 

3. 

26.  3000. 

27.  3s,  5^,  1& 

1.            28 

.  £189. 

29.  8. 

30.  25. 

21-1.  4/y.V)4    _ 

97/y.67i«^i3 

81«V/ 

x''y 

2. 

4. 

5. 

6. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 

17. 
18. 
19. 


32 

x"  +  ^x"  +  12a;  +  8.  3.  ic*  -  8.c^  +  24a;^-  32a;  +  16. 

x''  +  15x*  +  90a;'  +  270a;=  +  405a;  +  243. 
1  +  lOo!  +  40a;'  +  80a;'  +  80a;*  +  ^2x\ 
8w'  -  12w^  +  6m  - 1.  7.  81a!*  +  108a;' +  54a;' +  12a; +1. 

16a;*  -  32aa;'  +  24a'a;'  -  Sa\v  +  a\ 
243a;^  +  SlOax'  +  lOSOa^a;'  +  720aV  +  240a*a;  +  32a*. 
64a'  -  144a=&  +  108a&'  -  275'. 
a^x^  -  Zct^x^if  +  3aa;?/*  -y^, 
a'^x'^  +  4a'a;^  +  ^a/x^  +  4aa;'^  +  a;*. 
32a^m'  -  80a*/?i«  +  80a'w^  -  40a'm«  +  lOar^i'  -  'm}\ 
a'  -  3a'?>>  +  Za?c  +  3a5'  -  ^dbc  +  3ac'  -  5'  +  Zlyc  -  35c' +  o>. 
1  -  3a;  +  6a;'  -  7a;'  +  ^x"  -  Zx^  +  a;«. 
a'  +  3a'6a; + 3a  Q?  -v  ac)  X'  +  (6ac + 5')  5a;' + 3  (ac  +  7r)  ca;* + 35c'a;» 

+  c'a;^ 
1  +  4a;  +  lOa;'  +  16a;'  +  19a;*  +  Ux^  +  lOa;"  +  4a;^  +  x\ 
l+5a;  +  5a;'  -  lOo;'  -  15a;*  +  lla;^  +  15a;«  -  lOo;'  -  5xU5x'  -  x'\ 
1  -  6a;  +  15a;'  -  20a;'  +  15x*  -  Gx^  +  x\ 
(0) 


ANSWEES  TO   THE   EXAMPLES. 

20.  a*-8a^6 + 4a*c + 24a''^'-24a^5c  +  (ja'c^-Z2ah'  +  iSal'c-2ial(^ 
+  4ac^  +  165'-  326V  +  24&V  -  She*  +  c*, 

21.  1  +  lOaj  +  25x^-  40a;»  -  190a;*  +  92^»  +  570aj«  -  360a;^  -  67 5x* 
+  810a;''  -  243a;^°. 


22.  1.  1  +  2a;  +  Sa;'*  +  2a;^  +x\        2.  1  -  2a;  +  5x*  -  4a;'  +  Ax\ 

3.  9-12a;+10a;^-4a;'4-a;\        4.  a*-4a^h  +  l0a'h''-12ah^+9l\ 

5.  4aj'  +  92/'  +  162* -123-2/  +  lQxz-2ij/s, 

6.  9a''a;'  +  4&*2/'  +  ^^"^^^  +  12a5a;2/  +  Gacxz  +  Ahcyz, 

7.  1  -4«a;  +  2a-x''  +  4a  V  +  a\^*. 

8.  4a*-4a^-'ra'  +  4a  +  4. 

9.  1  -2a;  +  Sa;'*  -  4a;^  +  3a;*-2a;''  +  x\ 

10.  1  +  6a;  +  15a;'  +  20a;»  +  15a;*  +  Ca;'  +  x\ 

11.  x'-4x^  +  10a;* -4a;' -7a;'  +  24a;  +  16. 

12.  1  +  4a;  -  2a;'  -  4a;'  +  25a;*  -  24a;*  +  lOa;". 

13.  a'  --  4a'h  +  8^*6'  -  lOaW  +  8a'5*  -  4aV  +  h\ 

14.  a^  -  Sa'x + 28a V  -  56a'a!'  +  70a*a;*-  56a'a;'  +  28a'a;«-  ^az'+x^ 

15.  l-4a;  +  10a;'-16a;'  +  19a;*-16a;'^+  10a;«-4a;^  +  x\ 

16.  a^-4a'x  +  6aV-8aV  +  lla*a;*-8aV  +  6a'a;«-4flx'  +  a^. 


23. 


1. 

±  2a5'c',  ±  7a;'i/'2,  ±  10a*&V. 

2. 

3aa;'?/'        7a;?/'        Sa;^* 
"^     55     '  ^    8a  '  ^  4a6'  * 

3. 

a'a;'y      2a2/'      45'c'       6aZ)C* 
2    '        3a;'  ^     5a*  '        7     ' 

4. 

^2a;2/'    ^  3a5V      2a5'       2a;'2/ 
5a'  '        4a;*   '      c'    '  "^  3s»  ' 

24.  1.  2a;  +  2/,  5a  -  36,  5a;'  +  Sx^/. 
2.  7a5  -  a',  4a;?/  +  52/2,  5a'6c  +  c*. 


25.  1.  l  +  2a;+3a;'.  2.  3a;'  +  2a;+3.            3.  3a+25  +  c. 

4.  x''^xy+4i/\  5.  2a'-3a+4.            6.  4a;'-2a6  +  26'. 

7.  a;'-2a;'  +  3a;-4.  8.  3a-6  +  5c  +  ^.         9.  x^-2xhj-^2xy''-y\ 

10.  l-3a;+3a;'-a;3.  11.  2-3a-a'  +  2a'. 

12.  p  +  qx  ■¥  rx^  -i-  8X*.                     13.  1  -X,  a  -2. 

14.  2a- 36.  15.  a; -2.         16.  a -I. 

(C) 


ANSWERS   TO   THE   EXAMPLES. 

26.  1.  421,  347,  69.4,  737,  1046,  4321. 

2.  2082,  20.92,  1011,  20.22,  129.63. 

3.  3789,  75.78,  15.156.  8642,  2211. 

4.  4.164,  8328,  2568.2,  11307,  31230. 

5.  4.044,  8088,  5055,  6633,  15165. 

6.  1.5811,  44.721,  .54772,  .17320,  10.535,  .03331,  .06324^ 
.07071.       

27.  1.  «  -f  2y.  2.  a  -  3.     3.  ic  +  4.     4.  2a  -  3&. 

().  a  +  85.     6.  2x  -  7y,        7.  m-  Anx,       8.  ax  -  5lx. 

28.  1.  a^  +  2a+  1.  2.  x""  -  Ax  +  2,  3.  a'' -  ah  +  l\ 
4.  05-  -  Aax  +  4a\        5.  2a;-  +  Axy - 3y\      6.  x^-x''  +  x-l, 

7.  «  -  &  +  <?.  8.  1-  2.7;  +  3a;-  -4a;». 


29.  1.  21,  23,  25,  32,  4.7,  48,  64,  9.6. 

2.  114,  11.7,  125,  108,  1.41,  192. 

3.  2.34,  206,  3.84,  32.1,  282. 

4.  46.8,  936,  6.42,  1025,  1.284.    5.  1.357,  .5848,  .2154,  1.587. 

30.  1.  3a;^,  2ab\  AyhK  2.  ax,  a,  x. 

31.  1.  2a;-  (a-^x)\      2.  a;'  (a+a;)-.      3.  ah  (a-h)\      A.  2  (x-l). 

5.  X'  (x+l),  6.  2  (x+a).        7.  a'*  (x  i-l).        8.  3  (ax+2'). 


32.  1.  3a; -2. 
4.  Sx""  +  14a;- 

■15. 

2.  2a;  +  3. 
5,  4a; -5. 

3.  3a;  +  5. 

6.  a;^  +  2a;  -  3. 

33.  1.  a  +  ar. 

4.  2/  -  2. 
7.  3  (a;  +  3). 
10.  a  (a'  -  h'). 

2. 

5. 

8. 
11. 

a;-l. 

x-2a, 
x""  +  y\ 
x^  -  2xy  +  y' 

i 

3.  2  (a;'^  +  2a;  +  1), 
6.  a;  +  3. 
9.  a  (a  +  h). 
12.  a;^  +  4a^  +  4. 

34.  1.  2a;  +  3. 
4.  a;-l. 
7.  a;  +  3. 
10.  3a;'  -  2xy  4 

2.  3a; -2. 

5.  a;  -  3. 

8.  x""  -  3. 

11.  a;  (2.^^  + 

2xy- 

3.  So-  -  2. 
(S.  x-y. 
9.  5a;^  -  1. 
-2/').     12.  a;-l. 

35.  1.  \2a''h\  ZQ>x^y\  ax'y  -  axy'',  ah''  -  ad\ 
2.  120a*2>^  lOa'^i!^^  ISOOaV. 
^_        8.  6  (a''  -  ¥\  12a  (a'  - 1),  120xy  (x'  -  j/^). 
iH        4.  24a'5^  (a'^  -  ¥),  Z6xy'  (a;^  -  y'*). 

I 


36.1.^,^, 
2. 


ANSWERS  TO  THE   EXAMPLES. 
m  7a?        a'  -  Zab 


a     '    a^  '   3  (w- 2ar)  '   5a  '   26  (a  +  2V)  ' 


(a;  -  2^^)  '   w  +  7i '  a  +  6  +c  '     Ax-ly 
^     c        c  +  y      ex  +  d 

.  a;  - 1     a;*  +  «'     a*  +  a'J'  +  J*     a;'  -  5aj     a  -h 
a     '       a;*      '        a^  +  h"^       '     a;  +  5    '\  a  +  &  * 

5. r  .  6. .         /. .  8. ^-  . 

2a -Zx  a; +  4  7.^-2?/ 


2a  +  3a; '  '  x^^2x  +  1  '  bx"^-  Zxy  +  2y* ' 

-^     ba^  {a-^x)  -^    a;'+4a;+4  -^     a;'+a!-2 

a;  (a^  +  aa;+a;^)  a;'+a;  +  l  aj^'  +  oaj+o 

-  -   a;^  -  «aj  +  a^  ,  .  3aa;'  +  1 

15.  — — .  IG, 


4a^a;*+  2aa;^-l 


37.  1.  3a;~6+~-.,  a-2a;  +  — ,2a;4-6+-^ 

a;  +  4  a  +  X  x-o 

2a-3a;  +  --^ ,   12a;  +  3  +  --^- . 
oa-x  4a;  -  1 

a;(a!'-2a;  -  3)      a^+  2x^     x^  +  xy  +  y* 

a;-2        '«  +  2a;'         a;  +  a 


oo    1    ^^^>  ^^y?  ^^^      ^^^^^5   4Z>2/^,  3rt2' 


2. 


«6c         '  12a6o  ' 

40Z>^r7/,  45a?>^>c^  48a-62/^  50a^ry^ 

G0a=6^ 
aV-&V,  «7/^_+^V      ^^  ^  2r^a;  +  ar*.    a*  -  2aa;  +  a;* 

5^!_Z  ^^^iJ^L  Q   ^-^N  a  +  a;,   2« 


ao   1      ^*  +  ^'        3a'--a5  +  2Z>"      25a-20Z> 
*     •2(a  +  J)6'       G(a-2»)Z>     '  12       * 

^     «&       ^IjL^     ^llJ*'     a^-aJ  +  Z/'' 
'a-6'    a^-J«'    a^^^i^'    ~~^F^b^ ' 
2  g  +  ^a;     2a'^-2ah  +  2h^       2ah       x-y 

(8) 


ANSWERS  TO  THE  EXAMPLES. 
4  ^^    0       5  ^  •    6         ^  7    ?^? 


-.     «^  +  a;^  -^   2aj*+4a;y-22/*  a' +  a;' 

11.  —r— r-.  U.     T •  lO.     —- r  . 

a^  (a  +  a; j  x*-y*  _  ^^  (a;  -  a) 

y                       -  ,    a?  -  3a;^  +  3a;'                ^  .    1  +  2aj  +  3a;' 
14.—^—.  lo.  — -j -^— .  16.  —-m ir* 

'  b(a-x)'     '  Z> 

a*  -  a;*     a^  +  2a''a;  +  2fl5a;'  +  a;'     2ax'^  (x  -  y) 
a^x    '     {a-'X)  {a?  -  aa;  +  a;^)    '  c 

^       ab  X  a^x  (ax  - 1)  .    a'  +  6'      _    a;  - 1 

(a^-l'^)b      Za  x^'  +  b^ 

^'       a'        '2b'  ^'    x-b  • 


4- 3a;         3a;         6-2^  18a; -fl4       27 -4a;        12a;-40 
10    '    15 -2a;'     2a;+5'         21     '    2  (4a; -9)'     33-2a;  ' 

10- 13a;  20  -  3a;  14 -20a; 

6       '  ^     ^'  2a;-25'  9(a;+l)* 
1           J^      a^  +  a;'^  ,         _4      a;  (1  +  a;  +  a;') 


a^  +  5tf 

ace          ^     abc 
cd-be'      '  a  +  b' 

4. 

ace  +  Jc^  -  b'^e 
b{ae-c')     ' 

5.1 
c 

«4- 

7          ¥h 

''  a/-^2bc-bfg' 

Ua 

25  {a  +  1)' 

0.4. 

2.  f. 

10.  5^. 

43.  1.  1. 

3.  1.          4.  -  f . 

a'-a6  +  6'' 

6.1^. 

^    b'-a' 

|.        8.  -1.        9.  f 

• 

10.  -(a+c-b). 

11.  1. 

12.  Za, 

15.  2.          14.  - 

2^ 

'-^y   '»•-■!■ 

16.  8. 

17.  4. 

18.  Sf.           19. 

11 

20.  20. 

21.  8. 

22.  14. 

23.  -  107.      24. 

i. 

25.  -J. 

26.  0. 

27.-^. 

28.  H.           29. 

10, 

30.  4. 

(9) 


ANSWERS   TO   THE  EXAMPLES, 

44.  1.  144sq.yds.    2.  75  gals.     3.  £36,£1G,  £8.    4.  25  of  each. 
5.  £210.  C.  22.  7.  2450,  19G,  98.        8.  £200. 

9.  42,  GG,  1G2.         10.  C9,  81.        11.  £5  Ss.        12.  84. 
13.  15  ft.  by  11  ft.     14.  is  Sd,         15.  20  lb,  15  lb,  15  lb. 

16.  22,  £5.  17.  £48.  18.  3J  days.  19.  75.  20.  £125. 
21.  1504.  22.  1540,  880,  GIG.  23.  23»  days. 

24.  37i',  25'.    '     25.  7h.  5j\\  Gh.  IGyV-  2G.  13. 

27.  110  yds.      28.  £72,  £108.  29.  13 J  days,  2f  days. 

30.  £32.  31.  10  lbs.  32.  18,  lOi,  Gf  days.  33.  40'. 
34.  45  4itZ,  is  lO^d,  35.  £48,  £32,  £4,  £G5.  3G.  6§  oz. 
37.  30  hrs.  38.  40.  39.  90,  120.        40.  G54. 

41.  12  gals.  42.  25s,  20^.  43.  120,  104. 

44.  G2,  93, 155.        45.  76,  30.        46.  12,  21Js.      47.  36'. 
48.  21  p\  hrs,  lOiJ  hrs.  49.  189.  50.  1 J  hr. 

45.  1.  ic  =  1,  y  =  1.         2.  a;  =  -  6,  2/  =  a  +  J.         3.  a;  =  5,  y  =  2. 

.  a-  1)^  h  -  a* 

/.  x=l,  y-2.        8.  jr  =  — -    y  =.  —  9.  ic=6,  2/=7. 

10.  aj=l,  y=7.     11.  a;=10,  y=2i.  12.  ic-144,  y=216. 

13.  ic=  — ^~j — -■ ,    y=  — — — 7—  .       14.  ic  =  2,  2/  =x  3. 
/ir/7-LA/.      >    ^         ad  +  be  '  ^ 

^^•^-~^-2^-'2^=^-26- 


1 Q      _  ^^^  { J<5  -  6^  (J  +  c) }       _  «Jc  { h  (a  +  c)  -  ae\ 

19.  ic=G,  2/=8,   20.  ii'=3,y=2.  21.  a;=5,y=2.  22.  ir=-2,  y=-J. 
23.  a;- 7, 7/ =  9.        24.  aj=5,  y  =  5.        25.  a;  =  21,  y  =  20. ' 


46.   1.  a;  =  1,  y  =  2,  2  =  3.  2.  a;  =  7,  y  =  10,  2  =  9. 

3.  aj  =  5,  y  =  6.  2  =  7.  4.  a;  =  4,  y  =  -  5,  2  =  6. 
5.  a;  =  -  5,  y  =  6.  2  =  -  2. 

G.  x==i(l)  +  c-  rr),  y  =  i  (a  +  c  -  Z>),  2  =  ^  (a  +  J  -  c). 

7.  a;  =  1^,  y  =  2f,  2  =  -  12.  8.  a;  =  2,  y  =  -  3,  2  *  4. 

9.  a:  -  12.  y  =  12,  2  «  12.  10.  ic  =  5,  y  «  7,  e  «=  -  3. 

(10)  ~" 


ANSWERS   TO   THE  EXAMPLES. 


47.  l:  tV  2,  21, 40.  3.  5^,  3^.  4.  £24,  £12. 

5.  17  yds,  13  yds.  6.  48.  7.  108  sq  ft. 

8.  G40,  720,  840.  9.  18^,  90.  10.  4  hrs,  6  hrs. 

11.  75.         12.  40,  90.  13.  20,  30,  60.  14.  222. 

15.  30,  50,  and  70,  20  :  or  CO,  20,  and  40,  50. 

16.  24,  72.       17.  72,  60.        18.  12,  12.       19.  34.      20.  3L 
21.  12,  10.  22.  255.  23.  3^.        24.  39^,  2b,  12^. 


34^2  224  1  5 


a5^+  a^-va^h^  <-  ab^ ,  ah^  +  ab^  +  ah^+  ah\ 


a"- + 2  J ' + 3c'''+ 4a&^  +  ba'^l;  a^l}-^^  Za'h-  +  5^5^ + 4a-  »5  +  2a-»6»: 
12       3^     _5         13  5  42 

ia^h-^'c-^  +  4:a~^b-'c^  +  2ar'bc  +  \a-'b-'c-\ 
\abc^  +  ^aWc"  +  fa'^o'V?  +  5a-^Z»"*c; 
and 


3a-»6''c='      a'^Jc-^      ab-'c-'      'dabc  ' 
1  2  3  5 


3 


2a-'b-'c^      Za'^b-'^c-'^     Aa^h^C^      ab^c'^ 

V«  +  2  V«'  +  3  Va'  +  i\/a  +  ya\ 
\la       y^b     2V«?     V^     Vb^ 

\/b'  "■  2V(j  ""  3  v^^'  ""  TvTt  ■"  5Va'* 

5c_     ac_      1       <j'        1       v^      V^'      1 


c 
'a^b^ 


■  O    7  ^^'  7.  3      V^        V«'        V«'         ^' 


Z)c' 


V^'^  'y'b'  V« 


.*  .4 


49.  1.  ab'^,  a^  a"%  o'&.  2.  oj-^t/*,   oj^y^         3.  aj'y-2,  a^h^'e, 

4.  a;'»  +  6js*+9«^-4y.  5.  « -  &^  6.  a'-646^ 

T.  a''  -  a^  +  2a*  -  2  -  a"^  +  a"'.      8.  a? + ic^y '^  -  ic V*  -  y~\ 
9.  8aj*  +  4^5^^"^  +  2x^y  +  y^ ,  x'^  +  a;' V*  +  y"* 
10.  a"2-2a-«&*  +  4a'M-8a-^6  +  lGa'M-326^, 
ic?  4-  22?^  +  3^2f  ^  2x^  4^  1. 
(11) 


ANSWERS   TO   THE   EXAMPLES. 

11.  4a  -  2ah'^  +  2 aM  +  5"*  +  h'^c^  +  A 

12.  ah-'+  Sah-'+Zah^a-'l',  ^^xhj-'--2x+y--^^-x"y. 

13.  a'-  Ca^i^  +  2uhi  -UaJj^  +  G3a^6* - 54a^6^  +  27&. 

14.  «"* --  4«  +  106^^  -  16a^  +  10  -  I6a^  +  10a"^-4a-^+  a"l 

15.  X*  -  4a;^2/^  +  ^^'2/'  +  4.r V^''  +  ^ '") 

15  H  n  3     1_5.  ^  2-5 

a;-r  -  ^xhj2  +  lOaj^y'^  -  lOx^y  ^  +  5x^1/'"  ~  ^  ^  5 

a^h-'-iah-'  +  G  -  4a'-^6  +  a-''&^ 

^2^,-2  _ 5^2 j-2  +  lOa'^J  ^-  10^  ^2>^  +  5a  2^,2  «  ct  252, 

16.  ab-'  +  1  +  «-'&.  17.  a^  -  fa^  +  3  -  6a'K  9a"^. 
18.  a"*ic^- 1  +  a V*.        19.  iC2/"^  -  «"*2/^-       20.  2x2 .  Zy^. 


60.  1.  64*     8r^,  a)\  (#,  (SA  S\ 

2.  25*   (-#,   (K)*    (faO*,  {i(a'  +  2a&  f  5')}^; 

125^  (i|^)5-,  (^V«)t  (-V-^'A  U(«'+3a'&  +  3aZ)«+&«)i* 

6561"*,  (^-v^«-)"*,  K)'*,  (^'y*. 

4.  V125,  V3,  Vl'2,  V?5  V^,  V320. 

5.  V54,  V25G,  V2048,  ^3,  VI)  Vt^- 

6.  Via,  VOSa'-'o;,  ,  J    a/ -, ,    -^ 7 — Ti' 

*  /4^j2      »  /2fi       

8.  3V5,  5V5,  3GV3,  SyS,  18  «/2,  iV^,  V12,  VH  6. 

9.  4  V2,  8  V2,  C  V48,  f  V2,  /y  V2,  t  V2,  ?  V21,  I  V^SO,  V375. 
10.  2V3,  15 V3,  iV3,  AV^,  >V3,  W3. 

51.  1.  V108,  V112 ;  V81,  V^^  5  ^120,  V128,  V135 ;  V125,  V121  • 
Vi,  V-};  V125::^144,V1G2. 
2.  V2,  3V5.        3.  -2/  V3,  9  v'9.         4.  24v3,  120v3,  3G. 
21G  'yCj.  288 "V72.  C.  6-J(j,  Gv3+3v30. 

(;12) 


ANSWERS   TO   THE   EXAMPLES. 


10. 

11. 

52.  1. 
5. 
9. 

63.  1. 

6. 

64.  1. 

7. 

13. 

65.  1. 

4. 

66.  1. 
4. 

67.  1. 

'       4. 

68.  1. 
5. 
9. 

59.  1. 

5. 

60.  1. 
3. 
6. 


16.  8.  i  (V2  +  V3  +  V5),  W^  +  i  V32  +  i  VISO. 

J  (2V2  +  V3),  V5  +  1,  V5  -  V2,  4  +  V2,  i  (7  +  3 V5), 

^V(7V14-13). 

^V  (58  +  8  V7),  tV  (8  V5  +  23),  J  (3  -  V6). 

1 


(*  +  V«'-ic^    2Va'-a;^ 


4a;V^^-l. 


12.  2x\ 


l-x'' 


V3  +  1.  2.  3  +  V2.  3.  V5-V3- 

4V2~3.       6.  iV^-l-      7.  2-^v3. 
V2fl.        10.  V^-l.       11.  i(V5  +  l). 


4.  2V5-3V21 
8.  fV2-i. 
12.  V^+iV3. 


2. 


2+./^r 


3. 


2^^ 
2(^^))  • 


r^^2vs* 


4. 


7.  «'-5.    8.  T>.    9. 


26 


5-2' 
10. 


5.  a 

?>  (Z>-2a) 
'6b-2a  ' 


±2.        2.  ±3. 
±|.        8.  ±5. 


3.  ±1. 
9.  ±3. 


4.  ±i. 
10.  ±5. 


5.  ±i. 
11.  ±2.     12.  ±2. 


6.  ±2J. 


:.V3.    U.±%i    ^r.^  ,^ajc^l-nc.^^      ,,  (n-l)« 


2  (c^  +  d') 


16.'- 


V2?2-l 


4,  -2. 

7,  5. 

1,  -8. 
-1,  -12. 

6,  -5§. 
14, -lOf. 


2. 

-1. 

-9. 

5. 

8,  - 

-40. 

2. 

17, 

-4. 

5. 

1,  - 

-20. 

2. 

6,  - 

-4i. 

5. 

12, 

-12tV 

10,  2. 

If, 
2,  if. 


Ih 


2.  3,  -  1.  3.  2,  -  f 

G.  7,  -  LV.        7.  2,  i. 
10.  3,  -|.        11.  i(27±V57), 


11,  -13.      2.  5,  5f.  3.  5,  21. 

6,  3yV         6.  5,  -  41-.        7.  1,  10§. 


3.  20,  -  6. 
6.  10,  - 110. 

3.  -5,  -20. 
6.  25,  - 130. 

3.  8|,  -10. 
6.  13,  -  113j. 

8.  i(-9±3v3). 
12.  2,  -  3. 

4.7,  -1?. 
8.  3,-8^. 


aj^  -  4aj  -  21  =  0. 
lGaj*-153aj^  +  81  =  0. 


2.  6^^  +  5.2;  -  6  =  0. 

4.  iC*-6^'+lla;^-Gaj  =  0. 

6.  4a;*  +  3a;*-17ic'-12a!^+4x  =  0. 


ANSWEES   TO   THE   EXAMPLES. 

61.  1.  a;=7, 3/=  ±4.      2.  x=4.  y=-3, )         3.  x=4,  2/=3,  ] 

x=-S,y=4.]  x=^lVj,y=m\ 

4.  a;=8,  y=2i,        )    5.  ir:=6, 2/=5,     )         6.  ic=5,  y=3,       ) 
=  -8.^     iC=-6,  2/=-5.^  a;=J,y=-lJ.S 

=,3  ) 

«=i  A,  y=  -  t'i^.  )  x=  -  l-fV,  2/=  -2tV  5 

9.  a;=4,  y=2,>  10.  a;=10,  y-15,  ) 

fl;=2,y=4.5  x=-lOly=-l^.\ 


x=-2ly^ 
7.  ic=5,  y=,3  )  «.  a;=3,  ^=4,  ; 


11.  aj=3,  y=2,         )  12.  a;=5,  y=4, 
ic=  -  2,  y=  -  3.  ^  a;=4,  y=5.  ^ 

13.  a;=i{a±V26^£^},)  14.  x=i  {±  V^^Th^  +  h\,\ 
y=i\aT  V26-^-  a'},  t  y=  i  j±  VS^Tj'  -  5}. ) 

15.  a;=8,  y=l, 


l,y=8.i 


16.  x=  ± 


y- 


^=\y=^A  v^v;p'^      v^^ 


62.  1.  ±  12,  ±  15.  2.  i  10,  ±  16.            3.  ±  4,  ±  12. 

4.  15  yds,  25  yds.  *                5.  8  and  G,  or  56  and  -42. 

6.  27  yds.  7.  4550.                     8.  24  or  -  3. 

9.  4  or  -  1  J.  10.  40  yds  by  24.       11.  9,  12,  15. 

12.  12  and  7,  or  -9i  and  -14^.     13.  10  yds,  16  yds.     14.  3  in. 

15.  26  ft,  38  ft.  16.  16.         17.  49,  £3.          18.  4  ft,  5  ft. 

19.  £60  or  £40.  20.  10,  15.                   21.  £275,  £225. 

22.  25,  20.  23.  264.                       24.  2,  5,  8. 


63.  1.  aj  =  3, )     ic  =  23, )     a;  -  31,  2, )     a;  =  30,  15,  | 
y  =  lj     y=    2,f     y=    2,5,f     2/=  1,    8.r 
2.05  =  5,)     x=   5,)     05=49,)     05  =  11,)         33559, 

4.  a;  =  5,  y  =  3,  2  =  6.  5.  5.  6.  4,  2.  7.  4. 

•  8.  A  gives  14  pieces,  J5  9.     9.  8  ;  16.      10.  21,  12.       11.  4 
12.  59.         13.  8  h.  g.  and  3  h.  c.        14.  £13  Is  or  £42  1«. 

15.  By  paying  £5  and  receiving  4  louis. 

16.  3,  21,  16,  or  6,  2,  32.  17.  503. 

18.  2s,  45, 5s.  19.  209.  20.  301. 


64.  1.  32,  272.        2.  39,  400.  3.  63,  363.  4.  694,  3475a 

5.  9.  16.  6.  -1,0,  7.  -28.  8.  -275. 

9.  16i.  10.  -84^.  11.  336|.  12.  -84. 

(14) 


ANSWERS   TO   THE   EXA3IPLE3, 

65.  1. 12.  2.  5.  3.  20.  4.  -  ^^. 

5.  5,8,11,  14;  -2,-6,-10,  -14. 

6.  3§,  4i,  4f ,  5f ,  6,  61  7 1 7^,  8| ;  -11,  -9,  -7,-5,  -3,  -1,  + 1. 

7.  4,  15,  26,  37,  48,  59,  70,  81,  92,  103 ;  -  2i,  -  2^,  -  2i, 
-2,-11, -H,-li,-l. 

8.  -2,  -li,  -i,  i,  1,  If,  2i,  3i,4;  -2|,  -2^,  -1|,  -1, 
-la,  f,  lf,2.  9.  5,7,9.  10.  -3^,3^,10. 

11.  -  i,  i,  li.  12.  n\  13.  300. 

14.  78,  90.      15.  £5  35 ;  £135  4^.      16.  5  miles,  1300  yards. 


66.   1.  64,  85.      2.  1280,  1705, 
5.  4096,  3277.        6. 
9.  4AV 


3.  96,189.      4.  -256,-170. 


10.  2|ii. 


512,-341.        7.j\%, 


12.  72^V 


67.  1.  8, 

7. 


3 


2.  H. 
8.1. 


3.  J.         4.  A. 
9.  1/,.    10.  H. 


6.4. 
11.  lOJ.  12.  -2jV 


0.  ^. 


68.  1.  4. 


2.  3,  6,  12,  &c. 


3.  4,  -  8,  16,  &c. ;  or  — t  -  |,  -YS  &c.  4.  A- 

5.  3,  15,  75,  375  ;  or  -  2,  10,  -  50,  250. 

6.  ±4,  8,  ±16;  ±2,  8,  ±  32. 

7.  i -1,1,-1?;  -  1,  H, -j2J,  3f. 

8.  2  +  1  +  1+  &c.;  or4-$  +  |--&c. 
f-&c.  10.  l^or6J. 


9.  3-1  + 


69.  1.  -4,0),  4,  ...f,  f,  ^;i^,|f,lr^^,...15,-7i,-3; 
A,  J,  J,  ...  If,  2i,  31. 
2.  2|,3;li,f,JL,^^VV,rV  3.24. 


70.   1.  Si,  3,  2i«.  2.  2^,  2i,  2yV  3.  8  and  2.  ^ 

4.  1  or  16.  5.  8  and  2.  6.  9  and  1,  or  f  and  -  7J. 


71       1      JLS     1«     158     161.    Jt95         735        847 

4aJ 


2. 


X  +  y     x^  +  y^ 
^    x"  -  \\x  +  28 

6. r- 


(15) 


a  -1) 


A        4 


5. 


a'  +  d^x^  +  .t* 


:     7.1.      8.^;^^.      9. 

6^  (a  -  h) 


ad  —  be 


d* 


AKSWEES   TO   THE   EXAMPLES. 

72.   1.  10,4^,2^.        ^       2.  9,4i,iJ.  3.  G,  If,  1|. 

13.  (])  x  =  h  (-^7^)  ;  (ii)  x  =  a+hor  I  (a-l)  5  (iii)  a;=l,  y=4 

(iv)  a;  =  ±  9,  y  =  ±  3.  14.  3.  15.  25,  20. 

16.  8  : 7.  17.  £200,  £150.  18.  300. 

19.  £125,  £166f,  £2081;  £212J|,  £159^^,  £127J|.        20.  G. 


73.  1.  icy=i|(a;'+i/).     2.2.      3.y  =  ^-^.    4.  2/=3ir+2aj«+fl;». 
5.  y  =  a;^  +  2aj+3.        Q»:  z  = -^^x  +  lx\  S.  iACBC, 

74.  1.  720,  720.  2.  5040.  3.  6720,45360,  3326400,  19958400. 

4.  12600.     5.  9.    6.  1120,  831600,  336,  34G50. 

7.  6.   8.  7.   9.  15.   10.  3628800.   11.  0.   12.  4. 

75.  1.  126,  84,  36.   2.  330, 330,  11.   3.  3003,  455.   4.  6. 

5.  63.  6.  210,  84.  7.  50063860,  5006386.*  8.  18.  9.  12. 
10.  11.         11.  12.         12.  43092000. 


76,  1.  1  +  6a;  +  15a;'*  +  20a;*  +  15a;*  +  Gx""  +  x\ 

2.  a'  +  7a'x  +  21aV  +  35aV  +  35aV  +  21aV  +  7ax*  +  x\ 

3.  1  -  8a;  +  28a;'  -  56a;»  +  70a;*  -  56a;^  +  28a;°  -  8a;^  +  x\ 

4.  a'-9rt«a;+36<:iV-84aV+126aV-126aV+84aV-36aV 
+  9ax^  —  x^, 

6.  1  +  12a;  +  66x^  +  220a;' + 495a;* + 729a;* +  924a;«+792a;U495a;» 
+  220a;'  +  66a;^''  +  12a;'^  +  x^\ 

6.  1  -  20a;+180a;''  -  960.i;^  +  3360.?;*-8064a;*+ 13440a;'  -  15360a;^ 
+  11520a;»  -  5120a;'  +  1024a;^''. 

7.  a'  -  ISa'x  +  135aV  -  540aV  +  1215aV-1458«a;*+729a;«. 

8.  256a;«  +  1024ara;^  +  1792a'a;«  +  1792aV+ 1120a V+  448a V 
+  112aV  +  lOa^a;  +  a\ 

•9.  128a-^-1344a«a;+ 6048a  V-15120a*a;'+ 22680a  V-20412a'a;» 

+  10206aa;«-2187a;^ 
10.  l-5a;+V^'»-15a;'+i«^a;*~-V-a;*  +-'^a;«-i|a;U/3V^»--^3V» 

11. 1  -  -y-^  +  .^^x'  -  -Vx'  +  :jVV-Vt**''+i|4««-|i§x'+ jfi,*' 


(15) 


ANSWEK9   TO   THE   EXAMPLES. 

3.  1  -  (jx  +  27x^  -  lOSx^  +  405x'  -  &c. 

4.  1  +  Ca;  +  24^;^  +  80^=»  +  240aj*  +  &c. 

7.  1  +  x-\x''^\x^-\x^^k(^,  8.  l-2a;~a;'-|a;'-^a!*  -&c. 

9.  1  +  f  ic  +  -V-i»''  +  tI^'  +  tII^*  +  <^c. 

10.  1  -  Ix'  +  -V-aJ*  -  -||a;«  -  -^'j.aj^  -  &c. 

11.  1  +  la;  +  fa;^  +  ifa;*  +  -^i^x^  +  &c. 

12.  \-\x^  -^  -Ja;*  ~  -i^a;'  +  ^y^  ^'  -  &c. 


78.  1.  i  +  Ja;  +  f^a;'  +  ^^a;*  +  /^a;*  +  &c. 

3.  a^  —  oT^x  +  a'Wx"^  —  <^^5'a;^  +  a'^JV  —  &c. 

4.  <*^  +  2«  'J'a;  +  3a  *&V  +  4a^i V  +  ba'^b'x'  +  &c. 

5.  a^  +  6ah^  +  21a^&?  +  5Ga«5  +  126/^"p  +  &c. 

2  9  .13  23  r^n 

6.  a^  -  }a'^  aj'  -  2%a  ^  a;*  -  rl^a'"^'  ic*  -  AV^"^  •''^^  ~  ^^' 

7.  a-  Zah'^  +  6^^*r«  - lOa'^b'  +  15a^&'^ -  &c. 

8.  a^  -  ^a'^x  -  ^a'^  x^  -  -ijof^x^  -  i^^a'^'x'  -  &c. 

9.  a^  +  J^a-V  +  2\a-"aj^°  +  jy^a-''x'''+  ^^^a-^'x""' + &c. 

10.  a^  -  ^aJx""  +  f a"3a;*  +  /ya^^'aj"  +  -.h^"^'^^   :-  &c. 

11.  a' 2  +  !«"%«  +  ^cf^'x*  +  -^^a'^x^  -  lj||a""^'ic*  +  &c. 

_1   .1      .4  2      .15        10  8  13   11 

12.  a^x^  ^\a  %^  +  la  ^x^  +  \\a  ^x^  +  oV^a^^ic"^  +  &c. 


79,  1.  5221,  203116.  2.  100101100,  102010,  10230,  2200, 1220. 
3.  41104,  23420,  14641,  7571,  5954.     4.  235,  1465. 
5.  511,  22154.      6.  1212,  1212201. 


CO.  1.  1  +  14  +  244  -f  4344  +  114144  +  2050544  =  2214223  (sen.) 
=  mill  (den.). 

2.  100001000000  (bin.)  =  201000  (quat). 

3.  1756  X  345  =701746  (oct.),  1337  x  274=  381011  (non.), 
345,  274.         4.  57264,  95494,  e7^8. 

5.  4112,  6543,  62i^^.    6.  1295,  216;  2400,  343 ;  4095,  51? 


m 


MISCELLANEOUS  EXAMPLES:  Part  L 


1.  (a^  -  h^)  h^  +  (a»  -  Zah^  +  P)x-  {2a  -  I)  ax\ 

2.  3aj-  -  2al)x  -  2«^Z»^-  3.  3»-.  4.  (w  +  n)  a,  ^^f^^- 

^  x^  +  xy  +  y^ 

5.  lp\,  8.152.      G.  g^,    1^^.        r.  98,  in{Zn  +  25). 

8.  lyV  9.  5^5,  7aV27c,  ^4.         10.  l+a;+5a;^+|a;»  +  -V-a;*  +  &c. 

IL  (i)  ic  =  5  ;     (ii)  a;  =  5  or  -  1^ ;     (iii)  oj  =  4.  7/  =  3 ; 

(iv)  a;  =  ±  3,  y  =  ±  2,  or  ic  =  ±  2, 7/  ==  ±  3.  12.  |. 

13.  \-'r^x^2x''-bx^-x^+x''+lx\       14.  -2ai'^  +  8iry-5/.      15.  68. 

16.  X'  -  y\        17.  125,  1.709.        18.  x,  ^-^*.        19.  2v.,  2}. 

'x-y 

20.  a%^c^,  aj'^A     21.  a"^|l+a-^a;H-2a  V+-VVa:»+-V.^-V  +  «5cc.[. 

22.  1232,  11313,  363, 1044. 

23.  (i)cu  =  3f^;  (ii)aj=2or-f;  (iii)  ^  ==  5,  y  =  41. 

24.  12  days.   25.  x'  -  a\        26.  Ix"  -  5x-  +  Ix  +  9.   27.  f. 

9«   ^'      ^^-*-l        on  ^Ll?    2^ 

•  ic^  -  y'  4a^  +  2a  -  1*        5  -  a;'  ic^  -  1* 
30.  139,  1.39,  4.3955.   31.  2^%  .051.    32.  fljl  -  (  - 1)"(,  ||. 

33.  (rtic)"?jl  +  la-'x  +  i§«  =aj'  +  t%\«-V  +  ff|a-*a;^  +  &c.}. 

34.  7  ;  22  dollars  and  57  doubloons. 

35.  (i)ic=17;  (ii)  a;  =  60,  7/ =  40 ;  (iii)  a;  =  3  or  -  | J. 

36.  3f  hrs.        37.  20"!)^  +  2rrc^  +  2¥c'  -a* -I*-  c\ 
38.  1  -  ^aj  +  ^x'  +  ^f^^  -f  y^a;*  +  &c.        39.  {. 

40.  -^-^^'■4.       41.  139,  .6933.        42.  ^. 

43.  a'  {1- 7a V  +  4/-a V  +  icr'x'  +  Y'^V  +  &c.}. 

44.  i  Vm,  V^\  45.  93,  yV  (71^  +  n  -  6). 

46.  5221,  40255141,  6252711,  2451,  3341584,  1828. 

47.  (i)  a;  =  2J;  (ii)  a?  =  39,  y  =  21,  ^  =  12 ; 

(iii)  x=a . r,  y=:a .  -^-.     48.  64  days.    49.  a-h. 

a-Jy  a  +  h  ^ 

50.  9  +  1  +  49  =  59.      51.  3  {a"  +  J»  +  c')  -2(a&  +  ac  +  lc\ 

62.  a;'  -  9^^^-  53.  1054,  ^1  +  V2. 

(18) 


ANSWERS   TO   THE   EXAMPLES. 

^^'l^    ^y  ^5.  f|l-(^)«M.  56.6. 

57.  a^  {l  +  ^a-^^bx+^^a^b^x^  +  fi^a-'h^x^  +  i^^a  'h'x'  +  &c.\     58.  21 

59.  (i)  x=-i;  (ii)  x=Z,  y=i;  (iii)  a;=3,  y=l ;  (iv)  x=-l,  y=-^. 

GO.  16.  Gl.  2(Sx^y^-x^y-lx^  -  10y\        G2.  x^+2x  +  1. 

G3.  1.  G4:.  a'hUb -^  c\        G5.  2.C-1.        CG.  --]^%- . 

X  {4x^  -  1 ) 

67.  12.747.     C8.  (a^'x)'^^  {l  +  ia'x-^la-^x^  +  j^a'x'-^^%a*x'  +  &c,} 

69.  30.  70.  250,  G0300,  13874000. 

71.  (i)  oj  =  9 ;  (ii)  iz;  =  3^  or  -  4 ;  (iii)  x  =  -J-,  y  -  ^. 

72.  8  hrs;  17|  hrs,24  hrs,  40  hrs.  73.  G  (x  +  2x^  +  4/^  +  Sx-"). 

74.  |.      75.  aj,  -.-^"^  .       76.  55  ^7.        77.  1.772452,  x-i. 

79.  50f,  Jti (3/i+i).  80.  x' - 2x^-Sx^+Sx+1(j  =  0.  81.  28. 
82.  15120,  120.  83.  (i)  ic  =  1 ;  (ii)  x=2ly  =  ^;  (iii)  x=3. 
84.  £553 J,  £11061,  £3320.      85.  x'  +  l+  x-\  ^  8G.  a'  +  ax-2x\ 

87.  7.        88.  i  (1  -f  ^),  ?^-:!^ .         89.  Jx'''-Jxhh  hi 
'  36  -  2a 

90.  cc^  -1  -  x-i,  91.  ]-  71  (37i  +  1),  j\  { (I)"  -  1 } . 

92.  I  +x-^x^  +  liC* - -V^J*  +  &c.,  1  +  2a; - 2a;*  +  4a;' -  10a;*  +  &c. 

93.  27,  48.  94.  \  95.  (i)  a;=9 ;  (ii)  a;=4,  y=Z  ;  (iii)  a;=6  or  i. 
96.  4j\  miles  an  hour ;  134  minutes.  97.  x\ 

98.  a-  -  2ab  +  U\        99.  a;'  - 1.         100.  ?-^' , ^^^. 

^         ^x-'s/a'^y 

101.  4.11.      102.  l-«'  +  a;-^  +  Zcfix'i,  103.  7. 

105.  75,  25.     106.  V11333311  s^^jf.  =  2620  =  1000  den, 

107.  (i)  a?  =  9  ;  (ii)  a;  =  ±  -J  V^  5  ("i)  ^  =  "^i  V^^i   ^  =  ^• 

108.  £135,  £90.     109.  48.      111.  36a;*  -  97.t;*  +  36. 
112.  ^x'-'ax  +  \a\        113.  2.4494,  .4082,  .8164,  1.2247. 
114.  (ah-')Hm+i)^  1.   115.  a'-^^ah-Gac  +  AV  +  12^c  +  ^c\ 

4a^  -  2abx  -  («c  -  i?/)  .^;^  +  {^ad+ lie)  a;"-  -  (2&^Z-  ^c")  x'-  cdx* 
+  4cZV.  116.  2_5  |i  _  (|)"i^  81.  117.  -  11. 

118.  1  -  2a;  -  2.z;*  -  4.^'  -  10.i;*  -  &c. 

119.  (i)  a'-21 ;  (ii)  a;=  -  3  or  f  ;  (iii)  a;=5,  y=3.  120.  31   3. 

121.  8-12a^+18^^-27aT.       122.  aUla'hx-i(a'-h')x^+^ax^-ix\ 

S  r-  +  4x  +  2 
123.    Y-i ^ — ^.  124.  a^°-a°a;*-aV  +  a;^°. 

4x^  +  X  +  2 

125.    i.2247,  3  +  v3.  126.  c.  127.  0,  j\n  (7-7i}. 

12^    A  gives  26  guineas  and  receives  10  crowns. 
(19) 


ANSWERS  TO   THE   EXAMPLES. 

129.  2(a-x)  V2^,  i  \/a.  130.  33  :  238, 1  :  34. 

131.  (i)  2^=10 ;  (ii)  x=Z,  y=l ;  (iii)  x=^  or  -  1.  132.  IG. 

133.  With  upper  signs,  IG +9=5x5;  with  lo^cer^  0  +  25=5x5. 

134.  aj'+4?/.        13d.  x^  +  ax  +  h*  130.  -z —  ,    jr-r ^-. 

*^  ba  +  oy^    9a;-ic--3 

137.  mn  (w'  -  71^  (wi-  -  4/1^).  138.  a^  -  aj'^  +  1. 

139.  ^-^(a+h)  (a  +  c),      140.  4.8089,  .G803,  4.4494,  1.550C,  3.4494. 

141.  If,  2i ;  If,  If.         143.  (i)  a;  =  7 ;  (ii)  a;  =  4  ;  (iii)  a;  =  2  or  |i. 

142.  l+a;+|a;'+Y-^»-iJA;;*  +  &c.,  1  +  2j;  +  C.-c'  +  20.^'  +  70aj*  +  <S:c. 
144.  £9,  305.  145.  a'h'-ah'x-(a''  +  21')  x'  +  ax'  +  2x\ 
14G.  a^  -  «%'^  -  V-«^ + 2aM  +  4a;^  a*  -  2a^aj^  -  -^^a'^x + ^a 2a;^ 

+  iLiyiaV  -  23^^ic2  -  2Ga.>j'  +  IGa^oj^  +  IGa;^ 

147.  a;^-12-lG.r-S  a^  +  l  +  a-'.         148.  -^'^4-.     1^^-  -2154. 
'  a -26  + 3c 

150.  cc-2V^  +  l.  151.  V^-^V^.  152.  f  {l_(f)«l,l|. 

153.0,3,-2.        154.  G.        15G.  5idays,16days''.        157.  3j|. 

155.  (i)  x  =  —\:  (ii)  a;  =  f ;  (iii)  aj  =  ±  2,  y  =  ±  3. 

158.  a^  -  2ah^  +  zJb  -  2ah^  +  h\  159.  x-5. 

160.  ^^^^,  A.    IGl.  x'+2x'-Sx'-ex-l.         1G2.  9,  160. 
ic*  -  16       * 

163.  (&+c)^        164.  1147.        165.  l-6a:H24a;^-80aj«  +  240x«-&c., 
«^  {1  +  Za-'b  +  #a-^&^  -  la-»Z>»  +  ^a'*h*  -  &c.}. 

166.  33233344,  4344  =  1000  de?u,  244  =  100  deii. 

167.  (\)x=li  ;  (ii)  a?  =  -^-^-j ,  y  =  -r~i  5  (m)  aj  =  ±  G. 

168.  £40,  £28,  or  £28,  £52,  according  as  A  had  more  or  less  at 
first  tlian  K         169.  ^{a^  +  h^}  y(a'^h')^=Zl/2S9=10.SZ4, 

170.  X-'  -  4a;" V  +  3y\  171-  frl>~^^    ^-^i"  • 

1-0    1       1      a      o2  ^^r,     •   ax~h^  x  +  2 

1<2.  1  -i«a!-  -2^^^  1/3. ^- — jr.    -^  -  . 

(.^-a)(a;-?>)'    a;*-l 

174.  3.8729,  1.2909,  .'7745,  1.5491,  6.4549. 

175.  -10,  1/1(7- 3;?).  176.  15. 

177.  1  -  2x^  +  Zx  -  4xi  +  5a!'  -  &c ,  1  -  4a;^+ 10a;-20.T^+ 35a;'  -  &c. 

178.  12,  16,  18.        179.  (i)  ^~' ;  (ii)  2  or  -1 J ;  (iii)  a'=.49,  y =50. 

(20) 


ANSWERS   TO  THE   EXAMPLES. 

180.  10  days,  3 J  days.        181.  a» + ia''  (2x + y-z) + la  {2xy^2xz-^z) 
-  i^y^,  which  becomes  a^  +  Za^b  +  Sah^  +  ^',  by  putting 
X  =  h  =  iy  :^  -  iz,  or  x=  b,  y  =  2h,  z  =  -  2b, 

182.  ox''  +  laKi  -  -'^-a'h  +  ^a-'x^  +  Ja'^         183.  a;  +  1. 

184.    f  "^  ^-.        185.  «*a;  (a^ic*  -  1).        18G.  ah-'  -  ia^'b  +  !• 

X    +  X  —  o 

^^^-  (i^^T^^r  ^^^-  720,4(lfV7).  189.  4iff,|{U(-J)-}. 
190.  7h  37'  12".  191.  (i)  If ;  (ii)  x=4,  y=5 ;  (iii)  4}.  192.  10, 
193.  V'j.         194.  «'»  +  2.         195.  5  +  2  V6,  V^,  6  (5  +  2  V6). 

196.f!^^^^V,    ^1^4^.        197.  12x'-2..'-ll.'.l. 
\m  +  a  J  ^         a*  -  X* 

108.  §  {1  -  (-  f)"[,  1|.        199.  72,^        200.  ±  5a.        201.  15. 

202.  l-6^^-f21aj^-56aj+12Gaj^-&c.,  l-3^^+6a'l-10^+15aj^- &c. 

203.  (i)f;  (n)ar'ovb-';  (iii)  y  =  3,  y  =  1,  or  aj  =  f ,  y  =  J. 

204.  £800,  0.  206.  px^  +  qx-  r.  207.  a  -  ar'  +  4. 

208.^1^-^:1^^11^.     209.  --—1-—-.   210.7.0102,202. 
x-a  (ic^  +  l)(a;'+l)  ' 

211.  yV,  20.    212.  (aZ))T"2.  213.  4yds,  5  yds.   214.63361,236,34. 

215.  (i)4orl|;  (ii)  :c  = -5,  y=5  ;  (ii^)  x  ==-£=,  y  = -j^' 

^a'  +  b''  ^a^  +  b^ 

216.  -^^  days,  ^  days.  217.  2(n  +  4). 

218.  a^  +  j'^  +  c'  -  a^^  -  a^e  -  b%.        219.  2.        220.  x^  -  y*. 

221.  3.1622,  .12649,  2.1081,  1.5811,  4.4414,  .31622. 

222.  l{l-  {lY}^  2f.  223.  a'^  |l-3«- V+\-^a-V-^a-V 
+-^4^o^-V-&c  !,  i«-'  (1  +  3a-»ir+  ^^-^tt-V  +  V«~V 

+  -\Va"V  +  &c.!-. 
224.  £5825  85  5H  225.  0,  -  1,  2.  226.  20,  5. 

227.  (i)  .  =  «-or-i;  (ii)  .  = -^^_,  ^  = -_^^_  ; 

(iii)  ^=3  or  |.     228.    T?""  "" ''I  ^ays.     229.  2y'-ay-la\ 

230.  ic*  +  a-ic^  +  .t*,  a;'  +  2ax^  +  a'a;^  -  a\  x*  -  a^'x^ -'2a^x^  -  a\ 

231.  a"  -  b\      232.  x'  -  2.?j«  +  a;^  _  a;V  2a;  -  1.    233.  yV»  s~^-- 

234.  ioj^-  5yl  235.  88,  |  U  -  QTh  %  1}.  IJ,  1?. 

(21) 


ANSWERS  TO   THE   EXAMPLES. 

236.  la  ^  h,        237.  12,  4, 18  miles.         240.  ^'  ""  ^'  days. 

4W7l 

239.  (i)  x:=  ~6i ;  (ii)  x=  ^^- ,  y  =  , 5  (iii)  a;=10,  y=7,  «=3. 

241.  ic-  —xi/^+  x^y  ^  -y  *,  x^  -  (a  +  h)  X  +  ah. 

242.  3G,  125.  243.  5x  +  4.  244.  ^~,  J. 

245.  .8164,  1.6320,  2.0412,  .1010,  3.2549.         247.  3^,  3f,  4^,  «S:c. 

248.  7.  249.  720.  250.  248064^/60,  54373. 

n-i  ^N         -i-f     r-x  a*  +  ah  +  b"^  ah       ,.... 

2ol.  (,)  .:  =  17 ;  (n)  x  =  -^^^ '  ^  =  ^TTj '  <'">  ^  =  "• 

252.^^days,    r^!'^">   ,  day..     '   253.  140,VyVV 

43T21  11  2a— 6 

254.  ic'J  -x^y^  +  2/^,  a;^  -  (r/i  -  1)  «%*  +  a.     256.  1  J,  -,::7j- 

255.  18.r*  -  45aj»  +  ^7x'  - 10.^  +  6.         257.  ~--^,     J-^-, 

h  (h^  -  c')      V(l-a;^)» 

258.  24  miles,  i  hr.        250.  a^ji      260.3.      261,  a+2x:  a+Zx. 

262.  7425.        263.  (i)  cc  =  4  or  If ;  (ii)  ic  =  10,  y  =  -  3,  2  =  4. 

264.  Ss  id.       265.  -\'-  -  -^3^"-  =  Y  =  i  ><  t''2-  266.  A  +  2. 

267.  <z^  +  3a'^ic-  frt%'  +  2_via;'-^J^a"2  a;4  +  &c.,  a  +  6ic. 

268.  a.  260.  ^ ,    ,    ^  ,^^ , ^.  270.  £2  8^. 

{X  +  1)  (x  +  2)  (.V  +  3) 

271.  n  =  10  or  12,  Z  =  3  or  -  1.  272.  56,  44. 

273.  iji  (n-1)  (71-2).        274.  1111  x  10001=11111111=21845  deri, 

275.  (i).=CJ;(ii).=  ^^,y=  ^ZL^-;  (iii) x=4, 3^=3. 

270.  ^^P^:^"^  days.  277.  0.  279.  -^1^- . 

278.  ix64+  l  =  0  =  (ix  4+1)  (^x  16-i  x  4  x  1  +  1). 

280.  a^x^'-a'^x'^-a^xUl.      281.  xi-2x^-2.      282.  3^5,  -^V^. 
285.  30.  286.  15.  288.  4,  50,  65.        280.  12ah'c. 

287.  (i)x=  11;  (ii)a;=-4i;  (iii)  a;  =  4,  y  =  3,  or  a;  =  3,  y  =  4. 

200.  (-  ^1^  +  tV  -tIt)  -  (tJtj  +  2h  -U)  =  i  =  l2  (-ix  Jx  -  i). 

201.  1  -  V^a-V.        202.  84.         203.  2J,  .25208,  5  -  ^6. 

204.  («'  -Z,^)'.        296.  '-^  a,  -^^  ^ .        297.  10,  20. 
290.  (i)  (r=«c5-' ;  (ii)  a;=JV«;  ("0  «=1,  y=  -  1,  ora;  =  -  If,  y=?. 
208.  40320.  300.  ;  hr.  301.  3x^  -2.rV*  +  4y"^. 

(22) 


ANSWERS   TO   THE   EXAMPLES. 
302.  2.  303.  137G41,  a;  -  2  -  a;-^         304.  ft-tf. 

305.  V2,  V^  +  V2-  30G.   ^^, .  307.  1  hr  5j\\ 

X  —  y 

308.  ^V«'  {1  +  2cj'M  +  f^'%*  +  i^^a-'x  +  f?«'%^  +  &c.}. 

309.  1^1^71(7^  +  1);  f,^,l,  li.  3"l0.  6. 

311.  (i)  a;=100or  -10  ;  (ii)  x-  — v,  y-  -  -^;  (iii)  «  =  ,-^  . 
^  ^  ^  a  +  &  a  +  6  ^  1-^5 

y  =  fZ^^^  •  312.  I-  hr.  314.  11  h  or  5"^ 

313.  a  +  a"^;?;"^  +  a?,  x-2a^x^  -  a^x^  -  a, 

315.  Sahi-iJxy^+2a^x^y^-y^\     316.  -J  (V5-2),|x2^^+fa;  ^-5. 

317.  a^\l  +  ia-^x  +  -Ja-V  +  ^a'V  +  yV^a-'^c*  +  &c.}, 

J  {l-^a-'x  +  f^-V-^a-V+^-V^-V  -  &c.}.        318.  91. 
319.  mi-'  -11,  a-  \n  {n  +  1).       321.'£4  16^.       322.  63,  £62  8^. 
2al)  a''  +  hc  a"  +  he 

323.  (0^-—^;  {n)x  =  2',  (^u)x==-^,  2/ =  — — • 

324.  13,  12.  325.  1  -  x+\^x^  -  ^x'  +  -}x\  1  -2a;  +  V-^^  -  |a;« 

+  f  i^*  -  -6-^'  +  U^'  -  iTT^'  +  A^'-        226.  2.t;V'  -  3a? V. 
327.  2.64575,  .37796,  1.32287,  .88191,  1.47683.  328.  ii'K 

329.  2aV^:^  f,  V^^  330.  ^i^y^^- 

331.  l  +  2a;y-3ajy-^   332.  333.  7.  334.  27907200. 

335.  (i)x= or  -  1 ;  (ii)  a;=2,  y=l,  s  =  0;  but  indeterminate, 

if  2m  =  71  +  ^.  336.  5  miles  an  hour.         337.  c  or  c~* 

;8.     ^-fcl),  339,  4^„  340.  ^ 

341.  3  (ahc)i        342.  3.71,  1  -  2a;  +  3a:^        343.        I85.  4cZ. 

345.  (2a)^  {l-Aar^xi+  ^a-'x  +  ^\fl^"^a;"2  +  ^>-V  +  &c.;, 
(3<^)"^  {1  +  ^akx^  +  f|a"^a;^  +  fl«~'a;  +  |||a'M  +  &c.}. 

346.  2118760, 17296.      347.  (i)  x=a  ^ ;   (ii)  a;=l,  y=4,  g=27 ; 

349.  (x"  +  X  +  1)  a  -  (»  +  1).        S50.  a'^ -  a^e^.        351.  0. 
352.  #  -  |.,4  .  |.V  353.  4^-14^ 

354  ^til-d)-},  13i.  355.  7G.  356.  9  hrs. 

(23) 


ANSWERS   TO   THE   EXAMPLES. 
_n    3 


357.  (2ay^  {1  +  ^%^  +  5a-'x  +  ^a'ix^  +  ^Ya-'x*  +  &c.}, 
(3a)a  { 1  -  lah^  +  iah^ -  /^a-'aj  -  yf  ^a"^^^  -  &c. }. 

358.  4;  1023,  256.        360.  2f  hrs.        361.  x^i/^-x^yh. 

359.  (i)  ±  y  -^-— '- ;  (ii) ;  (iii)  x=-—  ,  y  =^  j—. 

f  c  c  a—0  o—a 

363.  «*  -  ic*  +  1.        364.  V^,  V2^.        365.  1.        366.  24«,  16a 

367.  a^  {I  +  2«-»a;  -  4«-V  +  ^^-a"  V  -  ^J^c^-^o;*  +  <fec.  f, 

a^  {1  +  4a-^a;  -  4a-V  +  %-«-  V  -  ^^a'^x'  +  &c.}. 
369.  71*.  370.  1023252  sen,  =  24^28  <?w<?af.  =  50,000  sq.  fU 

371.  (i)  a;=  -  (a-5  +  c)  ;  (ii)  ic=a  or  — ;  (ui)  x-±a  —  , 

a  na  ^^i  ^  ^a 

y=.±a  -p=--=--=: .     372.  lih.  Ih,  3h.     373.  0.    375.  tn^-^m^n^-l, 

374.  mx''-ax''^i'''X-(m-2)  a\     376.  ±  Vi^    377.  45.    378.  100,  4. 
379.  3Jhrs.      381.  15,  6.      382.  1,  7,  12,  or  2,  4,  14,  or  3,  1,  16. 

383.  (i)  aj=ll ;  (ii)  x=^  ^~^~ ;  (iii)  a;  =  ±  a,  y  =  t  &.        384.  £740. 

385.  (3.-2^)  x^.ia^-l^)  x,        387.  ^  J.        388.  ^~(|^^. 

386.  x^+(2y+l)  x^~(p''+p  +  l)x*-(p+l)  (2)^  +  l)x^+(p''+p+V)px^ 

+  (p  +  1) p^x-p^.        889.  rt      -a      ,avcb''^-vcf. 
390.  j\n  (3/1  +  1).        391.  2.549038  cub.  ft.        392.  12  hr  32y\'. 
393.  2f  394.  5880,  5880,  1960.  396.  2  gals,  14  gals. 

395.  (i)  a;  = -^-^^ 5    (u)  a;  =  i  (c  -  «) ;    0i03;=±256' 

398.  x  +  a.  399.  aj  +  6.  400.  icK  a;^  +  1  -  x^' 

iOl.  (a^h)^-'l.    402,  2x-\    40S,  5  (a-xy,m  {(a +xy-(n-l)  ax], 
404.  i(3  ±  V5),  Kl  ±  V^)-        405.  .985185312.        406.  215. 

407.  (i)5or-l;  (ii)  .a- or  c6- ;  (iii)  a:=  ±  ^--=,  y  =  ±-^^^, 

408.  4.  410.  2a;  +  Zy-z,  411.  293.9387.  412.  1-J.r^  +  a* 
414.  0, 1.  416.  \U,  418.  23.  420.  256  sq.  yds. 
419.  (i)  5 ;  (ii)  \a  or  Ja ;  (iii)  a;=0,  y=5,  «=(;,  or  a;=2a,  y^  -J,  a=  -<;. 

(24) 


